APPLIED FUNCTIONAL ANALYSISNumerical Methods, Wavelet Methods, and Image ProcessingABUL HASAN SIDDIQIKmg Fahd University of Petroleum & Minerals Dhahran. Saudi Arabia

MARCEL

~DEKKER

~

MARCEL DEKKER, INC.

NEW YORK BASEL

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PURE AND APPLIED MATHEMATICSA Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS

Earl 1. TaftRutgers University New Brunswick. New Jersey

Zuhair NashedUniversity ofCentral Florida Orlando, Florida

EDITORIAL BOARD

M. S. BaouendiUniversity ofCalifornia. San Diego

Anil Nerode Cornell University

Jane Cronin Rutgers UniversityJackK. Hale Georgia Institute ofTechnology

Donald Passman University of Wisconsin, MadisonFred S. Roberts

Rutgers UniversityDavid L. Russell

S. KobayashiUniversity ofCalifornia. Berkeley

Virginia Polytechnic Instituteand State University Walter Schempp

ManJin Marcus University ofCalifornia, Santa Barbara

Universitiit SiegenMark Teply University of Wisconsin.

w. S. Massey Yale University

Milwaukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS1. K. Yano, Integral Formulas in Riemannian Geometry (1970) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) 3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.: A. Uttlewood, lrans.) (1970) 4. B. N. Pshenichnyi, Necessary Conditioos for an Extremum (L. Neustadt, translation ed.: K. Makowski, lrans.) (1971) 5. L. Narid el al.. Functional Analysis and Valuation Theory (1971) 6. S. S. Passman, Infinite Group Rirgs (1971) 7. L. Domhoff, Group Representation Theory. Part A: Ordinary Representation Theory. Part B: Modular Representation Theory (1971,1972) 8. W. Boothby and G. L. Weiss, eds., Symmetric Spaces (1972) 9. Y. Malsushima, Differentiable ManWolds (E. T. Kobayashi, trans.) (1972) 10. L. E. Ward, Jr., Topology (1972) 11. A. Babakhanian, Cohomological Methods in Group Theory (1972) 12. R. Gilmer. Multiplicative Ideal Theory (1972) 13. J. Yeh, Stochastic Processes and the Wiener Integral (1973) 14 J. Banos-Nefo, Introduction 10 the Theory of Distributions (1973) 15. R. Larsen, Functional Analysis (1973) 16. K. Yanoand S. Ishihara, Tangenl and Cotargent Bundles (1973) 17. C. Proces;, Rings with Polynomial Identities (1973) 18. R. Hennann, Geometry, Physics, and Systems (1973) 19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) 20. J. Dieudonne, Introduction 10 the Theory of Formal Groups (1973) 21. I. Vaisman, Cohomology and Differential Forms (1973) 22. B.- Y. Chen, Geometry of Submanifolds (1973) 23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973,1975) 24. R. Larsen, Banach Algebras (1973) 25. R. O. Kujala and A. L. Vltter, eds., Value Distribution Theory: Part A: Part B: Defial and Bezout Estimates by Wilhelm Stoll (1973) 26. K. B. SloIal'Sky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The separable Galois Theory of Commutative Rings (1974) 28. B. R. McDonald, Finite Rings with Identity (1974) 29. J. Salake, Unear Algebra (S. Koh el al.. trans.) (1975) 30. J. S. Golan, Localization of Noncommutative Rings (1975) 31. G. Klambauer. Mathematical Analysis (1975) 32. M. K. Agoston, Algebraic Topology (1976) 33. K. R. Goodearl, Ring Theory (1976) 34. L. E. Mansfield, Unear Algebra with Geometric Applications (1976) 35. N. J. PuNman, Matrix Theory and Its Applications (1976) 36. B. R. McDonald, Geometric Algebra Over Local Rings (1976) 37. C. W. Groetsch, Generalized Inverses of Unear Operators (19n) 36. J. E. Kuczkowski and J. L. Gersting, Abslract Algebra (1977) 39. C. O. Christenson and W. L. Voxman, Aspects of Topology (1977) 40. M. Nagafa, Field Theory (1977) 41. R. L. Long, Algebraic Number Theory (1977) 42. W. F. Pfeffer, Integrals and Measures (1977) 43. R. L. Wheeden and A. Zygmund, Measure and Inte9ral (1977) 44. J. H. Curliss, Introduction to Functions of a Complex Variable (1978) 45. K. Hrbacek and T. Jech, Introduction to Set Theory (1978) 46. W. S. Massey, Homology and Cohomology Theory (1978) 47. M. Marcus, Introduction to Modem Algebra (1978) 48. E. C. Young, Vector and Tensor Analysis (1978) 49. S. B. Nadler, Jr., Hyperspaces of Sets (1978) SO. S. K. segal, Topics in Group Kings (1978) 51. A. C. M. van Rooij, Non-Archimedean Functional Analysis (1978) 52. L. Cotwin and R. Szczarba, Calculus in Vector Spaces (1979) 53. C. Sadosky, Interpolation of Operators and Singular Integrals (1979) 54. J. Cronin, Differential Equations (1980) 55. C. W. Groefsoh, Elements of Applicable Functional Analysis (1980)

56. I. Vaisman, Foundations of Three-Dimensional Eudidean Geometry (1980)57. H. I. Freedan, Deterministic Mathematical Models in Population Ecology (1980)

58. 59. 60. 61. 62. 63. 64. 65. 66.

67.68. 69. 70.71.

72.73.

74.75.

76.77. 78. 79. 80. 81.82.

83.84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 98. 97. 98. 99.100. 101.

102.103.

104.105.

106.107. 108. 109.

110. 111. 112.

n

S. B. Chae, Lebesgue Integration (1980) C. S. Rees el al., Theory and Applications of Fourier Analysis (1981) L. Nachbin, Introduction to Functional Analysis (R. M. Aron, trans.) (1981) G. Orzech and M. Orzech, Plane Algebraic Curves (1981) R. Johnsonbaugh and E. Pfaffenberger, Foundations of Mathematical Analysis (1981 ) W. L Voxman and R. H. Goelschel, Advanced calculus (1981) L J. Corwin and R. H. Szczarba, Multivariable Calculus (1982) V. I. Istr~escu, Introduction to Linear Operator Theory (1981) R. D. JalVinen, Finite and Infinite Dimensional Unear Spaces (1981) J. K Boom and P. E. Ehrlich, Global Lorentzian Geometry (1981) D. L Annacost, The Structure of Locally Compact Abelian Groups (1981) J. W. Brewer and M. K Smith, eds.. Emmy Noether: A Tribule (1981) K H. Kim, Boolean Mallix Theory and Applications (1982) T. W. W",ling, The Mathematical Theory 01 Chromatic Plane Ornamenls (1982) D. B.Gauld, Differential Topology (1982) R. L Faber, Foundations of Eudidean and Non-Euclidean Geornelry (1983) M. carme/i, Statistical Theory and Random Mallices (1983) J. H. calTOlh ef al., The Theory 01 Topological Semi9roups (1983) R. L. Faber, Differentiat Geomelry and Relativity Theory (1983) S. Barnett, Polynomials and Unear Conlrol Systems (1983) G. Katpilovsky, Commutative Group Algebras (1983) F. Van Oystaeyen and A. Versc!loren, Relative Invariants of Rings (1983) I. Vaisman, A Firsl Course in Differential Geomelry (1984) G. W. Swan, Applications 01 Optimal Control Theory in Biomedicine (1984) T. Pelrie and J. D. Randall, Transformation Groups on Manifolds (1984) K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) T. Albu and C. Nastasescu, Relative Finiteness in Module Theory (1984) K. HrlJacek and T. Jech, Introduction to Set Theory: Second Edition (1984) F. Van Oyslseyen and A. Verschoren, Relative Invarianls 01 Rings (1984) B. R. McDonald, Unear Algebra Over Commutative Rings (1984) M. Namba, Geometry of Projective Algebraic Curves (1984) G. F. Webb, Theory 01 NcnIinear Age-Dependent Population Dynamics (1985) M. R. Bremner et a/., Tables of Dominanl Weisj11 MultiPlicities for ReDresentations of Simple Ue Algebras (1985) A. E. Fekele, Real Unear Algebra (1985) S. B. Chae, Holomotphy and calculus in Ncrmed Spaces (1985) A. J. Jerri, Introduction to Integral Equations with Applications (1985) G. Katpilovsky, Projective Representations of Finite Groups (1985) L. Narid and E. Beckenstein, Topological Vector Spaces (1985) J. Weeks, The Shape 01 Space (1985) P. R. Gribik and K. O. Kortanek, Extremal Methods of Operations Research (1985) J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic AnalysiS (1986) G. D. Crown et aI., Abstract Algebra (1986) J. H. carruth ef aI., The Theory 01 Topological Semigroups, Volume 2 (1986) R. S. Doran and V. A. Belli, Characterizations 01 CO-Algebras (1986) M. W. Jeter, Mathematical Programming (1986) M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications (1986) A. Verschoren, Relative Invarianls 01 S/leaves (1987) R. A. Usmani, Applied Linear Algebra (1987) P. Blass and J. Lang, zariski Surfaces and Differential Equations in Characteristic p > 0(1987) J. A. Reneke el al.. Structured Hereditary Systems (1987) H. Busemann and B. B. Phadke, Spaces with Distinguished Geodesics (1987) R. Harle, Invertibility and Singularity for Bounded Unear Operators (1986) G. S. LOOele el al., Oscillation Theory of Differential Equations with Deviating Argumenls (1987) L. Dudkin et al., Iterative Aggregation Theory (1987) T. Okubo, Differential Geomelry(I987)

vv.

0, there exists a natural number N sucil that d(x",x) < {' for all n > N. In this case we write lim x" = x.

Definition 1.2.3

Q . ,>0erVf' t.bat elements of a metric space are pumts bm., in real life, we normally consider distances between points aud sets and between sets and sets, e.g., the distancf> hetweelJ a point and a linp in the plane I.md tbe distance between a square and a circle in tbe same plalle. This consideration motiwl.tes tbe study of Hausdorff metric

Hausdorff rVletricLet X bp a met.ric spaCt' with thp metric d and let H(X) delJot.e the set of all subsets of X For any two non-empty elements A. ami B of H(X), we define the distulJce between a point x in A and B by the relationd(:r,B) = int' {d(:J:,y)/y E B}. (1.2.1)

n

",,,. "'_'_""""'''''''10''-''

Very often, it is denoted by illL fl(x,?J).,cD

The Jist.unce bet.ween two elements uf H(X) is defined by tilt> relation

dCA, B) =

:,IUp

inf d(x,y)(1.2.2)

xEA yEO

= supd(x, B).

'oAItCUll

be shown by examples that

d(A, lJ)

oF d(B, A),

(1.2.3)

where

d(8,A) = sup inf d(y,x)yEn xEA

= sup int' d(x,y).yEfl XEA

(1.2.4)

Definition 1.2.6 Tile HmL.~dmJJ mctf"ie or the dist.ance bp.twell two elements A and B of If(S), denoted by h(A, B), is defined by the relatiun

h{A, lJ)Remark 1.2.4

~ max

{d( A. B), d( B, ,1)) .

(U.5)

if H (X) denotes tile set of all closed and bounded subsets of a metric space X, tben h(.4., B) is a metric. In particular, if 11 (n?) is tile set of all dosed and bounded subsets (compuct subsets) of It2, then h(A, B) is a metric.

Contraction Mapping or TransformationDefinition 1.2.7 (ContmcLion Mapping). A mapping T defined on a metric space (X,d) into it.self (T; X -) X) is calle.I a LilJSchit2 continuous mopping if there is a positi\'e real constant 0 such that d(Tx, Ty) < o{/( x, y) for all :r, y EX. If 0 < a < I, then T is called a co7ltmdiorl nWIJpirig. 0' is called the contmctivity factm> of T.Example 1.2.8 Let T ; R ---J. R ancl Tx = (I + X)I/3. ThClI finding a solution tu the equatiolJ T:r = x is \.-"C]uivalent to solving the equation x 3 - x - I = o. The mapping T is a contraction mapping 011 A = [I, 2J, where the contractivity factor is rx = (3)1/3 -1.

Q.

.... ,

, to is a",,"IO"--I

I,ormed space with the norm

III1 I = Sl~P I/(t)1 ,

[essential least upper bound

of I/(t)1 or essential supremum of I/(t)1 over the domain of definition of 11.

Example 2.2.15 The vector spaee BV[a,b] of all functions of boull(k'(l variatioll on [a,b] is a normed space with respect to the norm 1II11 = I/(a11+ 1;':- (x), where \ ';.:- (x) denotes the total variation of IU) on [a, bl. Example 2.2.16 The vector space Coo [a, &] of all infinitely differentiahk functions on [a, &1 is a nornwd sphce with rflSppct to tlw following norm:

II/II""

~

(

I. ~b "

)

ID'/(t)" 0 and center a which we denote by 5,.(a). This is nothing but M open interval (a - r, (l + r) . The closed sphere Sr( a) is the closed interval [a - r, a + r]. The open unit sphere is (-I, 1) and the do&.-',;I')JI(t 1,,-.. + I/C = U. If v E C[O,oo), then so is I and, in this case, the previous equation may be written as I = Tv where T is a line x E N.n-->:I;IProof of Theorem 2.1.1. We shall prove that (1) (3). (3) {::} (4) which means that all arc equivalent.}(2), (2) {::}]. (1) } (2): Suppose T is continuous, winch means that T is continuous at every point of X. Thus, (J) => (2) is obvious. Now suppose that T is continuous at the origin, that i (3). To pro\'e the converse, assume that T is bounded, i.e., there exists k > 0 such that IITxll -:=; k IIxll. It follows immediat.ely that if XII ---t 0, then TX n ---t O. Thus, (3) => (2).:1. (3)Q . ,~._ .._",,"l1>"-'......(4): Suppose l' is bounded, that is, there exists k > 0 such that IITxll < k IIxll. From this, it is clear that if lI:tll < 1, then lITxll < k. This means that the image of dosed unit sphere Slunder T is a bounded subset of Y. Thlls, (3) => (4). For the converse, assume that}TS I is a bounded subset of Y. This means that TS I is contained in aclosed sphere of radius k centered at the '0' of Y. If x = 0, thcn Tx = x and IITxll ~ k, and the result is proved. In case x =f. 0, IIxll E Sl and so l'(11:11) ~k orIITxll ~k 11:1:11 This shows that (4) :::} (3).IProof of Theorem 2.4.2. (a) (I) {::} (2): First, we pro\'e that if JlTII is gi\'en by (1), then it is equal to ;nf{kl IITxl1 ~ k IIxll1,II TII0'~'up { IITxl1 Ix l' 0 } IIxll40.IITII > IITxll, x _ IIxll ,mT,IITxll < IITllllxll a, TOThus,~0111'11isOllf'of th" k's satisfying the r"lat.ionIITxl1 ::; k Ilxli.Hence.1IT11 > mf {klllTxll < k IIxlllOn the other hand, for x f- 0 and for a k satisfying the relation k IIxll , we have III~IIII ::; k. This implies that(2.4.1)IITxll~IITxll sup { Wlxt'O }~k,or 111'11 < k. Since this relation is satisfied by all k and of :1: and k, we get111'11is indepcndent1IT11 < ;nf (kl IITxll ~ k IIxlllDy Equat.ions (2.4.2) and (2.4.3), we h1'l.v"(2.4.3)I!TII ~ ;nf{kl IITxll ~ k IIxlllThis proves that (1)~(2) ...... ,Q . ,>0 0, there exists a natural number N such that liT,., Tmll < (for all n,Jn > N. This implies that, for any fixed \'ector x E X, we have:l.IIT"x - Tmxll S II(T" - Tm )lIlkll N',"-->00that is, {T"x} is a Cauchy Sftluence in }'. 511lce Y is a Banach space, lim T"x = Tx, say. I'\ow, we \'erify that (a) T is a linear operator on Xinto Y, (b) T is bounded on X into Y, and (c) I)~, - TJI $"" for 7n > N. (a) Since T is defined for arbitrary x E X. it is all operator on X itJto 1'.T(x+y) = lim T,,(x+y)"-->00= lim [TnT l- Tnyl.. -->au ,,--> oc " --> 00as all Tn's are linear= lim T"x -/- lim T"y=T;,;+Ty T(ox) = lim T,,(ax) = lim oTnx sinlCe T,,'s are linear ,,--> ex, ,,--> ex, = 0 lim T"x = oTx.Q......."-->",,,, to 0 such that 1I1'"xll ~ AI for all n. This implies that for all n and any x E X, IIT'lxll < I)T"lIl1xJI ~ M Ilxli. Taking the limit. we have,,-->xHmIITn x11 < AI IIxll .IITxll < Ai IIxlllim ["-->00 Tnx=Tx. Since norm is a contitmolls functionn-'OOlim /IT"xllThis proves that T is bounded. (c) Since {T,,} is a Caudly sequcllce, for each t: > 0 t.here exists a posit.ivc integer N such that liT" - T",II < ( for all n,1II > N. Thus, we ha\'eIITxll.]IITnx-Tm < IITn -Tmllllxll lIxll xll0'fo' n.m> N."l~';".m Il liT - T",II = Slip {1I(T-T )X Ix oF U} ::; (, IIxll'Illfor all m> N;that is, T", ---t T as---t00.I2.4.3Unbounded OperatorsSuppose that T is an opcrator defined asExample 2.4. 17TfLet'YxooC'[O.I/.J"= sin nx. ThenIlfnll~sup Ifn(x)10 0 such t,hatIITxll > AI IIxllx' V, E X.The differential operator considered ill Example 2.4.fJ is IJIlbounded on ColO, II = {f E C [0, 11/ /(0) : /(1) = O} as well. However, it is bounded below on Co [0, 1] as follows: For / E C [0, L]I(:c) =1'df f -ds < Slip ~ o ds 0S"9 (IxIdorIITIII~supO 00,lIx n ll < IIxU n .which gives the desired result.Remmn 2.6.3For cvery element x of a Banach algebra,Verification. n = m. ThenThe result is dcarly true for n = 1. Let it be true forIIx"'+' II~IIx"'xll S IIx"'lIlIxll S IIxll'" IIxll ~ IIxll"'+' ,n.n". "'_'_""""'''''''10''-''or IIxnl+111 < Ilxll +' ; i.e., the result is true for m + 1, and thus by the principle of induction, the desired result is true for every n. Thcorcm 2.ti.l1f.1":I.~nlan element of a Banach algebm.\', Uten the sene$L x" is ,r:Ofl1JeT'!}entwheneverIlxll Xz ... :t: n E K, andare positive real numbers such that.LOi = I, theu.=1(2.7.2) Equation (2.7.2) is cal1ed Jensen's inequality. Sometimes this relation b taken as the dP-finit.ion of a convex functional. In view of Throrem 2.7.1, every affine functional IS convex. The converse of this may not be true. Every linear functional is COli vex.Example 2.7.2I. I(x) =x E (-1,1) f( 1) = f(l) = 2. The function defined in this manner is COIIVCX on [-1,1\.XZ2.I (x)= y2 is a convex function on (-00,00).3. I(x) = sin x is a convex function on [-r., 0].4. f(x) = Ixi is a eonvex fllnc.Uon on (-00,00) .5.I (x) =eJ" is a convex function on (-00, 00) .6. f(x) = -logx is a convex function on (0,00).7. I(x) = xl' for p > 1 and I(x) functionals on [0,00).-xl', for 0o ~ d 8 01 IXi-YilYil > o Vi. y,l 2'1+lx;d(x,y) ->0 - L 21+lxYI i=l .,-(Xl1IX; -ViiIf d(;I;, y) = 0, then~~lx, - Yil _ ~ 2' I + lx, -y,1 => => =>x0IXj-Vil -O~ vl J + IXi v.I IXi-Yil=OVi~IXi=::Vi Vi== (:tl, :(.2, ,xn , ... ) == V == (YI ,Y2,. ,Vn,"')" Conversely, if X ==y, tllell lx, -y,1 == OVi and d(J;,V) == O.n...... '"'_'_""""'''''''10''-''(ii) d(x,Y)~d(y,x).Verification.OC '"" 1 Ix, - y, d(x, V) ~ L 2' 1 + Ix _ y'=J.,I- :S'~~,''''\~ 1 1(-I)llx,-Y;1 2' 1 + I lllx; viily;-x,1 L2'I+lv -xl~ 1I 'd(y, x)X=(Xl.Xl, ... X" ... )(iii) d(x,y) < d(x,z) +d(z,y) wberey= (YI,U2, . . lUll"')Z=(21.22 ... 2 11 " , )Ver'ijication.1.d(x,V)~ ~f +lx,lx---z, ++z,_Y" I < f: I lx, - "I + ~ -'. 2'1+lx, zil1~2'1+lx;2' 1IIx,-y,1VII-;=1zIz - y , .=1~2il+lzi-y.1 .=1I>, - v;1by applying tlw ineCjllaJity given in Theorem D.3 of Appendix DThus,d(,,y) < d(x,z) +d("y).TheJ"efore, (X,d) is a metric space. This metric space cannot be a normcd space Lecause if there b a 1101'11I IHI such that d(x,Y) IIx ~ vII , t.hend(ox,oy) ~ II ox -oyll ~ lollix - vII ~ 101 d(x, V)n",,,. '"'_'_""""'''''''10''-''must be satisfied. But for the met.ric under consideration, this relation is not valid asd(ox, oy) = ::-: 2; 1 + 101andf-1101 lx, - y,l lx, y;I'101 d(x, y)2.~= I Ix; - v,1 101 L: 2' 1 + Ix--y- I . ~1=1 "(a) Ilxll~II" - y + vII S IIx - yll +IIyll, 0>' Ilxll - lIyll SIlx . yllAlso,lIyll - Ilxll < IIx - yll , -(llxll -llylD SIIx - vIITim!>, we get(2.8.1)(b) In (a), replace y by -y. Then, we getIIxll - II - yll S IIx + yllSince(2.8.2)11- yll ~ lIyll, IlIxll-ilylll S IIx + yll,which is the desired result.(2.8.3)3. In order to prove that is continuous, we need to show for f > 0, 3 0 0, such that II (x, y) - (x' ,1/)11 < f, whenever II (x, y) - (x', y')11 0, or equivalently> is continuous. Similarly, 1jJ is continuous as1I"'(o,x)-.,(~,x')1I ~ 1I0x-~x'II~llox-ox'+ox' ~x'il~lolllx x'il + 10 13111x'll aud so= 0 implies f(t) = 0 on [a, III] .(b) 1I0fil(e)=~ ( [ 10J('11' dt) 'I' ~ lal ( [ IN)I' d') >/, ~ loillfli. ~ (t If + g" dt) 'I' < ( [ ,fl' dt) 'I' + ([Igl' dt) 'I'IIf +gllby Minkowski's inequality (Th. D.5(b) of Appendix D). Thus, CIa, b] is a uorllled space as it is well known that C la, b] is a \'eCwr span> [Tn fRct, for operat.ions of addition and scalar multiplication, definedby11111 + Ilgll(f + g)(t)~f(t.) + g(t)(o!!(t) = of(t)the a.doms of vector space follow from the well-known properties of continuous fUllctions.] In order to show that C [a, bJ is not a Banach space with the illtegral Ilorm, WP I~onsider the following example: Take ft = -I, b == 1-1o N Equation (2.8.4) implies that(2.8.4)la:' - a;' I < {for n,m > N and Vi.(2.8.5)For fixl..-'l.i 1" Equation (2.8.5) implies that {an is a Cauchy sequence of real numbers and, by t1w Cauchy criterion of real sequences, it must converge, say, to (l.;; Le.,nnn =a; =a,(2.8.6)....... 0.::From Equation (2.8.4), we have, L la:' - a:" I>=1P< (I',for evcry k.Making m --+('X)in t.hisL la:' - /I,I;=),n~l;J\ion,we haveP< fP for n ;:::N.n.n","'_'_""""'''''''10''-''Making k --+00,we getLla~=-a,I P < (.P forn > N.,=JTillS implies t.hat- (:I;" - x) = x - x" Ell"where hence Furthermure,forn~N,x = (x - xn)+ x.,Etp .i.e. ,IIx n -xII--+ 0as n--+00.Therefore. {xn} is convergent to x E t p , which shows that t p is a Banach spacto. (c) c = {{J:,,} I lim '= is a veclor space with operations of addition and scalar multiplication defmed byn~=x" x}x+Y= (Xl +Yi,:I;2-I-Y2,. ,x.,+ lit,,.)ox:;::: (OXt.nx2,'.' ,0:1;", ... ),wherex =(Xi,X2, ...,x n , ... )11 = (lJl,m, ,.11",)belong to c and 0: is a scalar. For x E c, [Ixll = sup Ix"l. This is a norm. We shall now show that. c is a Banach space. Lpt. X tl = {ai'} be a CaudlY sequence in c. This implies that for (. > 0 3 N such thatIIx" - xmllThis implies that= sup lai';- a:JlI < (. for n. m > N.laj _a;lll < (. for n,m > N .(2.8.7)B . ,~,_ .._",,"l1>"-'......This !np.,ans that {an is a Cauchy sequence of real members and, by the Cauchy criterion, for fixed i, I lin a,m",-t:=aJ (2.8.8)By Equations (2.8.7) and (2.8.8), la, Since x" E c, we have lim ai'WE'Qbtain for n0.;1 ~ c,:=> N.(2-".9)a.. , which implies thatla" - ai'l Therefore, lam -akl:= o. ~1'(f) ~O; 1(0) ~O and II(I,}- I(t, ,)1~ 0 and:>~ 0for all lk {:} fUd = I(lk-l), and I(a) = 0I(td = O'rJk :> f = O.lIaIIi ~ 10/(0)1 + '';l' LlalJ(t,} - l(t,-d]1k=1"~ 101 [II(a)1 + "'P10111/11 E BV la,bj and h = I + g.~t,I/(I,} - l(tH )1]Let f,g Fot" any partition of t.he interval la, b] : a = to < /'1 < t'} < ... t" ~ b, Ih(a)1 ~ 1/(0) + g(a)1 O,Ij, #yIf(x) - f(y)l ~ K(I) < 00.Ix -ulShow that - is a subspiH:e of C [0, II. 2. Let Ilfll, ~ 'up If(t)1 + I 0 '"eh that IIT(x, 11) II < k 11>'11111111 Let1IT11Show I.hat~,uplI"'II:"O I ,lIyll"SlIT(x,y)lIIT(x,1I)11 < IITllllxllllyli.P7'Oblem 2.8.27 A linear operator T defined on a nOlllled space X UltO a normed space}' is called compact or (complelely contimlOlls) if, for every bounded subset AI of X, its image TUH) is relatively compact, i.e" the closure T(M) is compact.a. Show that t.he operator in Problem 2.8.20 i" compact.. b. Show that if 5 and T are compact operators, then 5 where a is scalar, are compact.+Tand aT,c. Show that every compact operator is bounded and hence continuous.Problem 2.8.28 Show that the identity opf'xator defined on an infinit.1 ~ IITII, then A is in the resolvent set ofT. Moreover, (AI - T)-' ;, g;ven by (AI - T)-' ~I:>-n-' T' and II (AI - Tj-' IIfl=O~S (IAI-IITII)-I . Show that the spectrum of T is contained in the circle P/P.I S limsuPIITfllll/"), which is contained in (A/IAI SIITID(e) Show that for a bounded linear operator T, u(T) is closed and p(T) is open.(f) Let T ; C[a, hI ---t C[a, hI be as in Problem 2.8.20, then (AI - T) -IiI = = v, A -=j: O. has a unique solution ii = A >. -"- T'lv.Ln~Problem 2.8.33 Inverse Operator Theorem; Let T be a bounded linear operator 011 a lIunne 1 is an inner product space with thf' inner(x,y) = LX;JJ"".=Jwherpx=(x\,X"l., . . ,xfl )ER f1 , !/=(YJ,Y2,.,y,,)ER".Example 3.2.3 Thp vprtor spa!'!' C [a, b] of the coutinuous function d{'-fined on [a, bl is an inner product SVRce with the inner productL (f,g) =lb[bI(x)g(x) dx, where I,g E C[a,bl2. (f,g) =tuI(t)g(t) w(t)dt, where f,g E C[a,b] and wet) 2: 0 belongsC[a,b].For wet) = 1, we get the inner product of (1). w(t) is called a \reight fUllction.Example 3.242 = {x = {x.d /n~l Ix.,lL x,y" .=J=< oo} is an inner productwhere x = {x,} E 2, Y =space with the inner vruduct (x,y) ={y;} E f 2 We caU .2 real if the sequences {x,,) Rre real.Example .1.2.!; L1.(a. b) = {All Lebesgue int.egrahl~ functions (real- or 2 complex-valued) over (a, b) / 1/1 are also Lehesgue integrable over (a, b)}is all inner product space with the inner product (f,g) =f,g E L,(a, b).lbIg dx whereRemm'k .1.2.21. An inner product space will be called finite-dimensional if the underlying vector space is finite-dimensional. Somet.imes 1\ finite-dimensional inner product space over the field of culllplex Ilumbers is called Hermitian space 01' unitary space. A finite--dimcnsionnl inner product space over the field of real numbers is called a Euclidean space.2. Unless explicitly mentioned, all abstract Hilbert spaces are defined over the field of complex numbers.Theorem 3.2.1 (Cauchy-Schwartz-Bunyakowski Inequality).all x, y belo1lgi1lg to a1l i1l1ler protluctSfKICCFol'X, me havel(x,Y)I' < (x,x)(y,y),(3.2.l)Proof. [f y = 0, then {x, y} = 0, by Remark 3.2.1 (6) nnd {V, y} = O. Thercfore, Equation (3.2.1) is satisfied as both sides are zero. Lt y f:- 0,then (y, y)f:-(x y) O. Let>' = ( , ). Then we havey, '!Jl(x,Y)I'(Y, y)~{:r., y){:z:, y} (y, y)A(x,y)~A(Y, x),by eunditiun (3) of Defillltiun 3.2.1. But w~ can writc >"{y,x) = X(x,y) by the same condition. Thus,l(x,Y)I'(y,Y )~Ay,x)~Ax,y)~((IAI' (y,y).(3.2.2)By using conditions (1) and (2) of Definition 3.2.1 and Remark 3.2.1(4) (a) and (b), we haveo'Y,x - >.y) = (x,x) + (x, ->..y) + {->"y,x} + (->.y, ->..y)-~ (x. x) - A(x, y) - A(Y, x)+ IAI, (y,y).(3.2.3)By Equation."-IThis is the desired inequality.ITheorem 3.2.2 Every inner protluct spacr X is a nonned space with resIled to the norm Ilxll = ](x.x)II/2 'r/x E X.Proof. Since the inner product space X is a veetor space by definit.ion, it suffices to verify axioms of t.he norm (Definition 3.2.1).1.0 'r/ x and IIxll = 0 if and only if x = o. Ilxll = [(J:, X)]1/2 , (x,x) ~ 0, and (x,x) = 0 if and only if x = 0 by condition (4) of the definition of t.he inner product. Therefore" (1) is satisfied.IIxll ~2.lIaxll= lalllxll 'r/ x E X and a real or complex. We have lIaxll = [(ux,ax)]IJ2 = [ao{x,x)]IJ2 by cundition (2) of Definition 3.2.1 and Remark 3.2.] (4) (b). Since an = lal 2 , we havplIaxll3.~ [ lal (X,X)']1/2~lall(x,x)] 1/2~InlllxllIIx T yll " IIxll + lIyll V x, y E X. [Ix+yll' ~~We have=+ (x,y) + (y,y) (x,x) + 2 Re{x,y) + (y,y)(x+y,x+y) (x,x) + (x,y)by Definition 3.2.] and Appendix D. By the Cauchy-Schwart'lrBunyakowski inequality, Re{x,y) < l{x,Y)1 < (X,x1/2 (y,y)1/2. Therefore, we hawIIx + YII' ]1/2 + [(y,y)]1/2f.(x, x)0'IIx + yll < [(x,x)I'I' + [(y,y)I'1' ~ Ilxll + lIyllHence, IIx + 1111 IIxl' + IIvll. This proves that norm on . X and (X, 11)1) is a normed space.sIIxll= [(X,X)]1/2 is aI0 . ,>0 N; i.e., lim (x n -Xon, X.,.,--too~0.......X on )::::: O ...,..-1"_""""'''''''10''-1Hilbert SpaceAn inner product space X is calk--d a Hilbert space if the normed space induced by the inner product is a Banach space (complete normed space). That is, every Cauch.y "equence {x n } E X with respect to the Horlll induced by the inner product is convergent with respect to thisnorlll.Definition 3.2.2Example 3.2,.6L R","-2and L2 la, b) are Hilbert spaces.2. C[a,b] and P[a,b] are inner product spaces but not Hilbert spaces.Note: We define an inner product on P [0,11 ns follows:(J,g) = wherej,gEP[O,l].l'f(t)g(t)dt,Example :Ut. 'lLet \" ={f/fis absolutely continuous on (a, b) withf and ~ belongingproduct:to 2(/, b) and f(a) = feb) = O}. Then Y is a denseSUbSlJilCe of Lz(a, b). Y is a Hilbert space with respt--'Ct to tlw following innerwhere (.,.) is the inner product of L 2 (a, b).Example 3.2.8 Let D be an upen 8uL:.et uf R~ and CO' (D), the infinitely differentiable real-valued functions wilh compact support in O. LetThen (...) is an inner product on CO' (0). This inner product induces the normQ...... ,, to abstract space is itself cal1etl Hilbert space in view of his valuable contributions in this area.2. In the early stage; of the devdopment of the Hilbert space theory, our present-day Hilbert space (Definition 3.2.2) was assumed to satisfy an additional condition, viz., the inner product space is also separable. That is, up to 1930. a Hilbert space was a separable complete inner product space.3. We can show that e...ery separable Hilbert space is isometric isomor-phic to '2.3.2.2Parallelogram La.w aIJd Cllaractelization of Hilbert Spa.ceFrom the elementary geometry (goometr.y of the plane), we know that the sum of the squares of the diagonals of a parallelogram is equal to thc sum of the sqUa1"es of its sides. The following result gives a. generalization of this result to an inner product space.Theorem 3.2.3 (Parallelogram Law).heloflging to an inne,' 1J1"oduct s]Jace. we have Fm' any two elements x and yIIx + yll' + IIx - gil' ~ 211xll' + 211YII' .Proof. Since X is an inner product. space, we have IIx + Yll2 = (x + Y, x +y) and Ilx - Yll2 = (x - y, x -V) (Theorem 3.2.2). By Definition 3.2.1 and Remark 3.2.1 (4), we ha.. . eIIx + yll' =0'(x,x)+ (x,y) + (y,x) + (y,y),(3.2.5)IIx + YII'Similarly,~IIxll' + (x, y) + (y, x) + lIyll' .(x, y) - (y, x)IIx . YII' ~ IIxll' -+ lIyll' .(3.2.6)n...". "'_'_""""'''''''10''-''By adding Equations (3.2.5) and (3.2.6), we obtainIIx + yll' + IIx - Yl"This is tllf' desired result.~211xll' + 211yll' .IRemar'k 3.2.6 The following exarnplf' shows that the parallelogram law is not valid for an arbitrary norm on a vector space. Let X = C (0, 21T] be the space of all real-valued continuous functions on 10, 21T]. Then C (0, 21T] is a Banach space with the normllfll = sup If(t)l. However, this norm does0"i + 2/!j + (>.. + 31!)k=> Xl =)" X"2 = 2ft,= ), + 31!Thus, a t.ypical element of A is of the form (XI,X2. Xl + ~X2). The orthogonal complement of A can be constructed as follows: Let x = (XI ..C2.X3) E A-.l. Then for y = (Yl,Y2,YS) E A. we have(.r., y) = Xl YI+ xzyz + XsYs =Xl YI+ xzyz + Xs(YI+ ~yz)= (Xl +XS)YI +(X2+~X1)Y2=0.Since YI and Y2 are linearly independent, we havf' xl+xa-OandQ...... ,, to"-IWe require the following results in the proof of Theorem 3.4.2.Lemma 3.4.1 Let At be a closed convex subset of a Hilbert space ).," (md p = inf lIyll. Then there exists a unique x E !If such that II.TII = p.yEAILemma 3.4.2 Let Ai be a closed subspace of a Hilbe7t space X, x fJ. M and let the distance between x, and AI be p, i.e., p = inf IIx - ull. Thcnthert'. e:l;isl.. O. (lfu = 0, then x -w = 0 and IIx - wll = 0 implies that p: 0.) Now, Wfe' show that ul..M For this, we show that for arbitrary Y E M, (u, y) = O. For any scalar a, we have lIu - O'1JII = IIx - IV - ayll = IIx - (w + ay)lI Since At is a subspace, w + ay E M whenever w, y E M. 2 Thus, w+ay E M implies that lIu - ayll > p = lIuli or lIu - ayll2 -lIull 2': 0 or (u-ay,u-uy) -lIull 2 > O. Sinc.;e (u-wJ,u-ay):::::: (u,n) -a(y,u)2 a(u, y) + aa(y, y) = lIuli - a(u, y) - a(y, u) + lal (y, V), we have,-a(fI.,y) - a(u,y)+ lal 2 11Y\l2 2': O.(3.4.4)By putting a ber, we get(3(u,y) in Equation (3.4.4), {3 being an arbitrary real num- 2~ I(u, y) I'+ ~' 1 y)I' lIyll' ~ o. (u,in EquatIOn (3.4.5), we obtain(3.4.5)If we put a =l(u,Y)1 2and b =IIYII 2-2{3a + {32 a b ~ 0,{Ja ({3b -2) > 0'rjreal {3.(3.4.0)If a > 0, Equation (3AG) is false for all sufficiently small positi\ ~ R. Hence, 2 a must be zero, i.e.. a = l{u,y)1 = 0 01' (u,y) = 0 'rIy E Af. IProof of Lemma 3.4.4. It is a well-knO\vn result of vector spaces t.hat AI +N is a subspace of X. We show that it is dosed, i.e., every limit point. of AI + N belongs to it. Let z be an arbitrary limit point of /rl + N. Then there exists a sequence {z,,} of points of M + N such t.hat z.., -t z. (See Theorem C.2 of Appendix C.) !lll..N implies that M n N = {OJ . So, every z" E A:f + N can be written uniquely in the form z" = ;1;" + 1/'" where x" E /l;f and y" E N. By the Pythagorean theorem for elements (x", - x,,) and (Yrn - y,,), we have(3.4.7)(It is dear that (x.." - x n ) l-(Ym - Yn) 'rim, n). Since {Z'l} is convergent, it is a Canchy sequence and so liz", - z..,11 2 -t O. Hence, from EquatIon (3.4.7),n...... "'_'_""""'''''''10''-''wc s(' that Ilx m- x,,11 ~ 0 and !lyon - Y"II ~ 0 as m,n ~ 00. Hence, {x m } and {V,,} are Cauchy se00lim Zn = lim (x n ,,---too+ Yn)= hm Xn"->00+ lim"->00y=x+yEAI+NThis proves that an arbitrary limit point of M is closed. I+Nbelongs to it and so itProof of Theorem 3.4.2. By Theorem 3.3.1llf..J.. is also a dosed subspace of X. By choosing N = Ml. in Lemma 3.4.4, we find that At + All. is a closed subspace of X. First, we want to show that X = AI + Ml.. Let X f. AI + All.; Le. AI + All. is a proper closed subspace of X. Then hy Lemma 3.4.3, there exists a nOlw..ero vector u such that u.l..M + Ai l.. This implies that (u, x + Y) = 0, Vx E AI and y E At1.. If we choose y = 0, thcn (u, x) = 0, Vx E M; i.e., U E Ml... On the other hand, if we choose x = 0, then (u,y) = 0 for all Y E M1.; i.e., u E Ml...l.. (Since M and Ml.. are sub-spaces this choice is possible). Thus" U E MoL n M1..l.. By Theorem 3.3.1 for A = M1., we obtain that u = O. This is a contradiction as u =I- O. Hence, our assumption is false and X = 111 + llll... In \'iew of Remru-k 3.4.1 (2), the theorem is proved. I Sometimes, the following 1>tatement is also added in the statement of the projection theorem. "Let Jf be a closed subspacc of a Hilbert space X, t.hen M = AI.L.L (Unsolved Problem 3.10.3)".Remm'k 3.4.31. Let.X == L 2 (-I, 1). Then X = Mffi 1If.L, where AI =~{fEL2 \ -1,1)1 f( -t) = f(t)Vt E (-1,1); i.e., the space of even functions};{f E L,(-I,I)/f(-t) odd functions}.M~ ~-f(tN' E (-1,1), i.e., 'he space of2. Let X = Ldo,bJ. For t; E [1t,bJ, let M =: {f E L 2 [o,b]/f(t) = 0 almost pverY\"'here in (a,e)); and JIl. = {f E L 2 [o,bJlf(t) = 0 almost everywhere in (e, b)}. Then X = AI ffi All...n...... "'_'_""""'''''''10''-''Remm'k 3.4.5 The orthogonal decomposition of a. Hilbert space, i.e., Theorem 3.4.2, has proved (Illite llseful in potential theory (Weyl, [17]). The applications of the results concerning orthogonal decomposition of Hilbert spaces can be found in spectral decomposition theorems which deal with the representation of operators 011 Hilbert spaces. For example, for a bounded self-adjoint operator, T, (Tot"y) is represented by an ordinary lliemanu-Stieltjes integral. For details, see Kreyszig [9), Naylor and Sell [II J .LetX=Lz [O,7I"1 andY= {fEL2 [O,7I"lffisconstant}. It can he checked t.hat. Y is a closed subspace of X and hence itself a Hilbert. space. Every h E ~ [0,71"] can be decomposed asExample 3.4.1h = f -I- g,f E V and 9 E VJ. (By Projection Theorem).For pxamplp, 11 = sin x can bp decomposPd as sin x = fl -I- (sin x - a), a E 1'. The constant n is determined from the orthogonality of 9 = sin x - n to every element e in Y(e,,q) =I' 1 e{sinx 0a)dx = U= e(2 - (m).2 Since c is arbitrary, we have n = -. Thus. til(> projection of h = sinx on Y,is the functIOn f(x) = -. Let us define an operator P on Lz [0,71"1 bynPh~ P(f+g)~f,2fEYPg =SlllX -(l9 E 1;.!..,-2I' 2 (10 (sinx-7I")'!tlx )'1' I" 4 1 [( 1 (sin'! x ..... j'T2 - 11" sin,r:}dx )]'1'0( 2"IIsinx - ell -,-I- :; - 71"48)1/2::::: 0,545[1~ (sin z x ...... ,] _ 2CSinX)tlot-] II'![; -I-c27f _ 4C]I/Zn...... "'_'_""""'''''''10''-''3.5Projection on Convex SetsWe discuss here the concept" of projection and projection operat.or on U)[lvex ~t.s which are of vital importane p in such diverse fields as opt.imization, optimal control and variational inequalities.Definition 3.5.1 (a) Let X be a Hilbert space and /( c X a non-empty dosed conveJ\ set. For x E X, by projection of x on [(, we mean the element z E K denoted by PIr) such thatIIx - PK (x)lI x projection of x on 1'11I(u, v) =" j-, bxdx = ["X2'] '_, = 0[J=2 ifu=v=l. The functIOnsu = nj lIull =I/V2 and v = vi IIvll = /fx are orthonormaLDefinition 3.6.21. All orthogonal set of vectors {1}i.} in an inner product space' X iscalled an orthogonal basis if for any u EX. there exist scalars such that000;u = L:Oit/!;.;=1If clements VIi are orthonormal. then it is called an orthononnal basis. 2. All orthonormal basis {'1ft;} in a Hilbert space X is called maximal or complete if there is no unit vector Wo in X such that {t/Jo, WI, 2,' .. } is an orthonormal set In other words, the sequence {W.} in X is complete if and only if t.he onl~' vectol' ort.hogonal to each of Wi'S is the null vector. 3. Let {wd be an orthonormal basis in a Hilbert space X, then the numbers 0; = (u, t/J,) are called the Fourier coefficient.s of the element u with rcsjwct. to thf' system {,pol and 'L,o;t/!, is called the Fourier series of the element u .B......, to O.u- L{U,\C'i)tP;=,L.H.S.NIlull' - 2 Li-I(u, ",,)(u, ,p,)NN+ LL(u,>!,.)(u,,pj)(v,,,,pj),=1 j=1by using properties or inner pluduct,~lIull' -L l(u.,p;)I'1=/Nn...", "'_'_""""'''''''10''-''by applying th fact that('I/J., tP)}This gives us= 0,) = 0 if i= I iri=j.i= jL l(u,.,,)I' < lIull'.=]NTaking limit N --+00.we getL 1(. ",)1' I'..=100(3.6.6)(rl) Any subspace V of X that contams {1p;}IS(lense m X.Proof of Theorem 3.6.4. Uniqueness of (3.6.3) is a consequence of Theorem 3.6.3(2). For any U E Y, we can writeu = Jim ""' o.,t!,;,/'.'-->00;=1L.J"M >Nas Y is closed. From Theorems 3.4.2 and 3.6.3, it follows thatIIMAiV -B.......II, to"-Iand as N --+00,we get the desired result.IProof or Theorem 3.6.5. (a) => (b). This follows from (3.6.3) and the fact t.hat {'I/J;} is orthononnal. (b) => (c). Put tt = v in (3.6.5) t.v get (3.6.6). (a) => (d). The statement ("-I(d) Since u.LN, u 2 0' lIull , thenr;. N; let;r= u E X - N. If f(u) = (u,au) =Therefore, for a =Hon of the theorem in this case, t hat is, /(11) = (u, au). (e) Since u11,,11f(u~, the vector y =0 And soau satisfies the condi-~N, f(u}-#~~:~is defined for any x E X Then f(x - Bu)Consider x - Bu. where B= ~~:~.= f(x)-(3/(u) = O. This implies that:l: - (3u E N. Every x E X can be written as x = x - f3u + Bu. Therefore. for each x EX, I(x) f(x - ~u + ~u) ~ I(x - ~,,) + I(~u) ~ I(x - ~u) + ~/(u) (f i, Uncar) = (x - (3u, au) + !3{u, au}~0'by (c) and (d), wh~ref(x -/1u)= --:rI(u)lIuli(. Smce x - (3u E N, hy (c),I(u) = --, ) .= (x -Bu,au) for every a and so for aThis gives us /(x) = (x - {3u,au} + ({3u,au) = (x, au). Thus, for all arbitrary x EX, there exists a vector y 'au such thai I(x) ~ (x.y). 2 We lIOW sbuw that y is unique. Let y\ and Y2 L~ twu vectors such that f(.1:) = (X,1/') Vx E X ('Ind J(x) = (X,Y2) Vx E X. Then (x,y\) =:: (X,Y2) Vx E X or (x,y, - Y2) = 0 '!>(x)) ~ (x,z) ~ (z,x).(3.7.3)Prom the definition of the natural imbedding J and Thcorem 3.7.2, (3.7.4) From Equauons (3.7.3) and (3.7.4), we have It = F which is the desired result. IQ .,_,_......,..",,"IO"-I.... ,3.83.8.1Operators in Hilbert SpaceAc{ioint of Bounded Linear Uperators on a Hilbert SpaceLet. X be a Hilbert space ,:IIld T : X --t X be a boundt--'d linear operator 011 X int.o itself. Thp-I! the adjoint oJ.!~mtor T* is defined by(Tx, y) = (x, T*y) V :/', Y E X.Definition 3.8.1Remark 3.8.1L T always exists.Verification./y(XILet (Tx,y):::: fy(x).'&2) = (T(xl + X2),Y) = (TXI ,V) -\- (TX2,Y) = fY(XI) + /y(X2) J,(ox) ~ (T(ox),y) ~ a(Tx,y) ~ oJ,(x).-l-Now, 11,,(x)1 = I{Tx,Y)1 < IITxU 1I11J1 (by Cauchy-Schw 0 such that IIT(T'x)1IHence, 0: Le., TS is positive. 2. (TT"x,x):::: (T"x,T*x}:::: liT xII'] > 0:::> is positive. Similarly, 2 (TOTx , x) :::: (Tx, T~ J as YO' =: T. Hence, (T*Tx, x) :::: IITxll 2: 0; i.e., TOT is positivc. ITrProof of Theorem 3.8.6. To prove the c1osedne&l, we 8how that the limit of the SffjUence of normal operators is a normal operator. Lct lim T" = T. Then lim T; = T*. 11-+= '1-->00IITT* - T*TII = IITT* - T'1T~+ T'1T.: - T.:T'1 + T~T'1 - TTII < IITT' - T,.T~II + IIT,.T; - T;T"II + IIT~Tn - TTII -+ 0a8n-tooB..... ,, to'1' IIxll ~ 11'11 => 1>'1 ~ l. I3.8.3Adjoillt of all UIlboUIldedLiIle~lrOpemtvrIn Section 3.8.2, WI" have studif'd the cOllcept of the adjoint of a bounded linear operator on a Hilbert space. This concept may he extended to unbounded linear opewtors on a BaJlach space, in general and on a Hilbert space, in particular. From the point of view of applications, the concept of adjoint of unbounded linear operators on Hilbert spaces is more useful and, therefore, we discuss it here in brief.Definition 3.8.4 Let T be an unbounded linear operator on a Hilbert spac~ X and assume that the domain uf T, V(T), is dense in X. The adjointoperator of T, T*, is defined by(Tx,y) = (x,T*y) for all x E V(T),lJ E V(TO),where V(T*) = {y E X/(Tx,y} = (x,z) for some z E X and all x E V(T)}. For eac.h such y E V(T), the adjoint 0lwral.or To of T is defillf~d in tR.rms of that z by z = TOlJ.Remark 3.8.4 If T is also closed, then V(TO) is dense in X and To is dosed. Morcuver, T*' = T; that is, V(T**) = V(T) and the operators agree on these domains. For dosed operators, see Section 4.7.Exarnple 3.8.10 andLet X = L 2 (a,b) and Tb~defined by (Tf)(t) =~V(T) = {J E X / f is absolutely continuous with~E X and J(a) =o} .0 . ,>0d [(,)_/9 d' (1) : (Tx,lI) == a(x, y) == a(y. x) == "{T"'lIcc == (x. Ty). This shows . x"} that T* == T that T is self-adjoint, I Proof of Corollary 3.9.3. Since T is self-adjoint, we can define a bounded bilinear form a(x,],) == (Tx,y) == (x, Ty}. By Corollary 3.9.2 and Theorem 3.9.1,IITII~lIall~IIFII== sup IF(x)1IITII11"'11=1 == sup I(Tx.x)1 11"'11=1 ~ ,up I(Tx,x}l. 11"'11=1IThe following theorem, known as the Lax-tvlilgram lemma proved by PD Lax and AN Milgram in 1954 [10], has important applications in different fields. Theorem 3.9.4 (Lax-Milgmm Lemma). Let X be a HilIlCrt s]Jace aud a(,) ; X x X --t R a bouuded bilinear form which is coercive Of' X -elliptic iu the se71se that then, exists a > 0 such that a(x, x) > 0' IIxlI~ V x EX. Also, let f : X --t R be a bounded linear fUflctional. Then then? exists a unique element x E X such thata(;c,y) == f(JJ) "IV E X.(3.9.b)Proof of Theorem 3.9.4. M > 0 sl1c;h t.hat.Since a(,) is bounded, there exists a constant~ MI" (v, v)1IIvll IIvll(3.9.9)Q . ,>0.Tu - )"Jv.v) = 0 ' 0, t.he affine mapping for v E X, V --t V - p()"'Tv - >'/y) E X is a contraction mappinp;. For this, we observe thatIIv -p..\Tvll:.! = (v - pATv, v - pATv)= dvll2 -2p{)"Tv, v)2fX' IIvll'+ p2 11ATvll 2(by applying inlier product axioms),; Ilvll' -+ p' Ai' IIvll' ,a(v,v) = (>.Tv,v) ~ allvlland2(by the coercivity),(3.9.12)II>.Tull ~ AI IIvll2(bv boundedness of T).Therefore, (3.9.13)IIv-pATvll< (1-2fX,+P'Ai')1/'1IIvll(3.9.14)Let Sv = v - P (ATv - A/y ). ThenIISv - Swll~II (v -P (ATv - Af,, - (w - P (ATw - Af,,)11~ lI(v - w) - p(AT(v -w)11(3.9.15)~ (1-2pa+p'M,),/2l1v-wll (by 3.9.14).n",,,. "'_'_""""'''''''10''-''This implies that S which is equivalent Banach contraction fixpd point. which isis a contraction mapping if 0 < 1 - 2pa + p2 M 2 < 1 to the condition t.hat. p E (0,20:/ !lI 2 ) . Hence, by the fixed point theorem (Theorem 1.2.1), S has a unique t.he uniqup solut.ion. IThe following problem is known as the abstract variational problem:Problem 3.9.1Find an plement x such thata(x,y) = I(ll)for aU y E X.where a(x,y) andI are as in Theorem 3.9.4.In view of the Lax-tl'liIgram lemma, the abstract wl.riational problem has a unique solution. Note: A detailed and comprehensive description of the Hilbert space theory and its applicat.ions is p,iven by HelmherK [8], Schechter [12] and Weidmann [16].3.103.10.1ProblemsSolved ProblemsIf X ::j;. fO} is a Hilbert space, show thatProblem 3.10.1IIxll =Solution.x::j;. O. Then,up Ilyll=1l(x,Y)IIf x = 0, the result is dearly true as both sides will be O. LetIIxll=WIIxll'(x,x)=Ilxll= (x,TIXfi)xsup (X,l/) lIyll=1 ~ sup IIxll lIyll, (by the Cauchy-Schwartz-Bunyakowski inequality) 111111=1=J.' ohLain1 IIlf - f,,11 < Ilf - f",11 + Ilf" - f",11 < 2P-' + 2P + 2P < 2"-" This shows that {/... 1 converges to 1 E .z (a, b) and hence 2 (a, b)a. Hilbert space.4. C [a, b] is an Inner product space but not a l-lilbert space: Foris1,9 ECIn, b],we define t.he inner product b.\, (I, g) =Iabf(t.)g(t.) dt.-It can be seen, as in the case of 2 (a,b) , that (.,.) salisfiL"S the axioms of the inner product. vVe nOW show that it is not complet.e with respect to the norm induced by this inner product, i.e., there exists a Cauchy sequence in C [a, b], whidl is Ilot convergent in CIa, b] . Leta=-I,b=l,andfor for for-1!~,up 1I-IIVxll'l.Therefore, the condition that III - VVII conditions.(4.2.7)E,-y -p ( T-xo ) ')= >+ xf(Y),asIf is linear, i.e.,-1) (-y T-;1:0)1 < 1'+ >..f(Y).(4.2.8)rVlultiplying (4.1.8) by>., we have(the inequality in Equation (4.2.8) is reversed as >. is negative), 0'Since-t > 0, by condition (2) of Theorem 4.2.1, we ha\'eand so(-),) (-DQ......, to(1 Problem 2.8.5). By Theorem 4.2.1, there exists F extending I which is linear fmd F(x) < p(x) 'V x E .\". We have -F(x) == F(-x) as F is linear, and so by the abo\'e relatioll, -F(x) < p( -x) ~ IIJIlIl - xII ~ IIJllllxll ~ PIX). Tim" IF(x)1 < PIx) ~ 1I/1111xll which implies that F is bounded andIIFII~II x ll=1,up IFlx)1 < IIJII(4.2.9)On the other hand, for x EM,I/(x)1 :::: IF(x)1 < I)FlllJxll, and sollxll=!IIJII ~ ,upIJ(x)1 ~IIFII(4.2.10)Hem:p. by Equations (4.2.9) and (1.2.10),11/11:::: IIFII.IRemm'k 4.2.1 Tlw Hahn-Banach Theorem is also valid for 110rmed spaces defin(..>(1 OYer the complex field.1.2.2COllsequences of tlJ Extension Form of tll Hll/m-BaIUldl TheoremThe proof.__ of t he following important results mainly depend on Theorem 1.2.2:n...", "'_'_""""'''''''10''-''Theorem 4.2.3 Let w be a 1l0mero vector in a normed space X. 'Then fhere exists a contifltlOus linear functional F, defilled on the entire slJace X, such that IIFI! = I and F(w) = IJwll. Theorem 4.2.4 fhen w = O.If X is a normed space such fhat F(w) = 0 V F EX',Theorem 4.2.5 Let X be a nonned space and M its closed subslmcc. FtIT'I.her, assume tJwt wE X - M(w E X but w /Il). Then fhen~ exists FE X' such Ihat F(m) = OfomUm E At. and F(w) = l. Theorem 4.2.6 Let X be a nonned SlJace. M its subspace and w E X stIch thai. d = inf IIw - mil > O. "'EM (It may be observed that this condition is satisfie ~ > O. Thus, I is bounded. n:::: W implies that), :::: I, and therefore 1(10) :::: I. 11 :::: m E M implies that), :::: 0 (see Equation (4.2.11)). Therefore, from the definition of I, I(m) =... O. Thus, I is bounded lineal' and satisfies the conditions I(w) :::: 1 and I(m) :::: O. Hence, by Theorem 4.2.2, there exists F defined over X such that F is an extension of I, and F is bounded linear. i.e., FE X. F(w):::: 1 and F(m) :::: 0 for allm EM.IAImEMII--J:!- - wll : :-:$IInll/d.IPl'Oof of Theorell1 4.2.ti. Let N be the subspace spanned by M and w (see Equation (4.2.11)). Define I on N as I(n) ::::),d ProCet.>(ling exactly as in the proof of Theorem 4.2.5. we can show that I is lineal' and bounded on N[I/(n)/:::: /),Id Since II(n)1 r V x E 5,. If we choose l' = 1, then by the above fact, there exists some closed set 51 such that S\ c S and an integer n SUdl that lIT'l1(x)1I > 1 V xES. F\trther, we can assume thatdiaJlleterufS 1 , Ii(SI) < 1. We repeat theabm'e procedurt> for S\ instead of S, and 7> = 2. Thus, we get a closed set S2 C S. SUdl thaI. Ii{S2) < 1/2 SUdl that for some 712, IIT"2(x)1I > 2 V x E S2. Continuing this process, v"e get a sequence of closed sets {Sj} satisfying tht> conditions of Theorem C.2(7) of Appendix C, and hence by this theorem, there exists some point y E nSf which implies that !IT,,.,(Y)II > i for everyi.Q . ,>0d if we show that xED and y = Tx. . The gh'en condition implies that (xn,Tx n ) -+ (x,y) E GT. Since G T is dosed, G T = GT; i.e., (x,1}) E G T . By the definition OfGT,y = Tx,x E D.1. In some situations. the above characterization of the dosed linear operator is quite useful. 2. If D is a closed subspace of a normed space X and l' is a continuous linear operator defined on D into another normed space V", then T is closed (Since x" -+ x and l' is continuous, Tx" -+ Tx which implies that if TX n -+ '!.J. then 1'x = y and morelwer. xED as D is a closed subspace ofRemark 4.7.2X.)3. A closed linear operator need not be bounded. (The condition under which a closed linear operator is continuous is given by the Closed-Graph Theorem.) fbI' this, we consider X = Y = e[O, 11 with sup norm (see Example 2.2.9). and P[O.I] the normed space of polynomials (see Example 2.2.10) We know that prO, I] C C[O, 1]. .Let T . D = pro, 1J -+ [0,11 be defined df as 1'1 = dt (a) T is a linear operator; see Example 2.4.\l (b) Tis dosed; let In E P[O,I].,,---toolimIn = I TI=and lim'1---+=dl"dl= g. To prove thatTis dosed, we need to showthatdldt= g. (It may be observed that the convergence is uniform.)We ha,'eQ......., to, it.5 inver.~e 1'-1 is atdomaticaUy contimwus.4.7.3The Closed-Graph TheoremTheorem 4.7.4 (Closed-Graph Theorem). If X and Yare Banadl spaCl:lj, and if l' is a li1tl:ar 01Jf.->f'ator of X into F, then T is continuous if and only its graph G T i~ closed.Remark 4.7.3 In view of the following result of topology, the theorem will be proved if we show that l' is continuous whenever GT is closed'Iff is a continuous mapping of a topological space X. inton Hausdorlf space F, then the graph of the product X x y'fis a closed subset ofProof of Theorem 4.7.4. Let XI be the space X renormed by IIxlll = IIxll+ IIT(x)lI S, ; " nocmed space. Since IIT(xlll S IIxll +IIT(xlll ~ IIxlh, T is continuous as a mapping of XI into Y. We shall obtain the desired result if we show that X and Xl have the same topology. Since IIxll :5: IIxII + 1I1'(x)1I = IIxlh, the identity mapping of Xl onto Y is continuous. If we can show that XI is complete then, by Theorem 4.7.3, thi~ mapping is a homeomorphism and, in turn,.-\ and Xl have the sanlf> topology. To show that XI is complete, let {x,,} be a Cauchy sequence lllit. This implies that {x n } and {1'(x n ,} are also Cauchy sequences in X and Y', respectively. Since X and )' are Banach spaces, there exist vectors x and y in X and F, respectively, such that IIxn - xII ~ 0 and 1I1'(x.,) - yll --t 0 as n --t 00. The assumption that G T is closed in X x Y implies that (x,Y) lies on G'r and so 1'(x) = y. FUrthermore,IIx" - xiI. = IIx" - xII + IIT(x. - x)1I ~ IIx. - xII + IIT(x.l - T(x)1I ~ IIx. - xII + IIT(x") - yll -+ 0 as nTherefore, XI IS complete.-+00.IB...... ,, to haw> 1'(x + y) = lim 1'.., (x--->00+ y).Since P&h Tn is linear,1;,(x + y) ~ Tn(x)n ---+00lim 1'.. (x --r y) ==~+ Tn(y) lim [1'n(x) + Tn(Y)] n---toolim 1',,(x)n---+=+ lim"'-+00T,,(y)T(x)+ T(y),0,T(x+ y)~T(x)+ T(y).(b)T(ax) = lim 1',,(03:)n~oo= lim ul~,(x)n~ooas each Tn is linearT{ax) =Qn~oolim 1'n (3:) = uT(x).Thufl, T is lineRI'. 2. Since lim l~(x) = 1'(x)n~ooII 10m Tn(xHI = IIT(x)lI,n~ooas a norm is a continuous function and T is continuous if and only if x" --t x implies TXt} --t 1'3: (see Proposition 2.4.1(3).n.n",'"'_'_""""'''''''10''-''Thus, llTn (x)l1 is a bounded sequenCf' in Y By Theorem 4.6.1 (the principle of unifonn boundedness), 111',,11 is a bounded sequence in the space B[X,Y]. This implies thatIIT(x)1I ~ lim IITn (x)1I < lim in! 1I1~lIl1xll.n ---> =,,---> ""(4.8.1)By the definition of111'11, we get 111'11 :S: lim inf 111',,11n~=(see Appendix 0.4 forlim inf and Definit.ion (2.4.1(6)) for 111'11.) In view of ~luation (4.8.1), T is bounded and hence contmuous as l' is linear.Problem 4.8.2 Let X and Y be two Banach spaces and {T,,} a sequence of continuous linear operators. Then the limit Tx = lim Tn(x) exists for ,,-->= every x E X if and only if1.IITnll:S:M for n = 1,2,3,n~=2. The limit Tx = lim Tn(x) exists for every element x belonging to a dense subset of X.Sol1ttiou. Suppose that the limit Tx = lim T,,(x) exists for all x E X. n~= Then clearly Tx ==- lim T,,(x) exists for x belonging to a dense subset ofX."-->00IlTn l1 < p..,for n = 1,2,3, ... , follows from Problem 4.8.1.n~=Suppose (1) and (2) hold; then we want to prove that T(x) ==- liml~(x)exists, Le., for > 0, there exists an N such that 1I1~(x) - T(x)11 < ( for n> N. Let A be a dense subset of X, then for arbitrary x E X, we can find x' E A such thatIIx - x'il < 0,0 > 0We hayeIITn(x) - T(x)1I " IIT,,(x) - Tn (x')11and arbitrary.(4.8.2)+ IITn(x')II- T(x')I1> N,(4.8.3)By condition (2),IITn (x') - T(x') II N.This pro\"es the desired result. Show that the principle of uniform boundedness is not valid if X is only a normed space.Pf'Oblem4.8.3Sohttion. To show this, consider the following example. Let Y = Rand X be the normed space of all polynomialsx = x(t) =with the normL u"t'ln=O~where an = 0 V n > N,IIxli = Stlp lanl.X is not a Banach space. Define T., . X ---t Yas follows:"Tn(x) ~,,-,La,IIT..II4.8.2 Unsolved Problems4.8.4k=Ois not. hounded.Pf'OblemProve the Hahn-Banach Theorem for a Hilbert spaceShow that p(x) = limsupx n , where x = (x n ) E: too,x n real and satisfies conditions (1) and (2) of Theorem 4.2.1.Problem 4.8.5[f p is a functional satisfying conditions (1) and (2) of Theorem 4.2.1, show that 1)(0) = 0 and 1)(-x) ~ -p(x).Problem 4.8.6n.n". "'_'_""""'''''''10''--'Problem 4.8.7 [f F(x) = F(y) for every bounded linear functional F on a normed space X, sho"" that x = y. P7'Oblem 4.8.8 A linear function F defined on ing conditions, is called a Banach limit:Tn,satisfying the follow-1. F(x)~Uifx=(Xl,X2,X:j,.. ,xn, ... )andxn >O'r/n.3. P(x) = [;fx=(I,I,I, ... ).Show that lnninfx n S; F(x) S; limsupx" V x = (x n ) E Banach limitTn,where F is aProblem 4.8.9 Let Tn = An, where the operator A: .2 --t .2 is defined by A(XI,X2,X3,X4, ... ) = (X3,X4,"')' Findn ---+ =Hm IITn (x)lI, IITn ll and,,---+ 00lim11~1I.Problem 4.8.10 Prove that the uormed space prO. 1\ of all polynomials with noml defined by Ilxli = suplO'II, where 01,0'2, . . are the coefficients,of x, is not completeP7'Oblem 4.8.11 Show that in a Hilbert space X, a sequence {xn} is weakly cOIl\'ergcnt to x if and only (x n , z) converges to (x, z) for all z E X P7'Obiem 4.8.12 vergence in norm. Problem 4.8.13 flexive.Show that weak convergence in 1 is equivalent to con-Show that all finite-dimensional normed spaces are re-B......., to0T(x+ l}t) 7]T(x)-D1')1 =, 0 lxfor every t E X, where 1] ---t 0 in H. DT(.-r:)t E V is called the value uf the Gateaux dcn:vativc of T at x in the direction t, and T is said to be Giiteau.?: differentiable at x in the direction t. Thus, the Gateaux derivative of an operator T is itself an operator often denoted by DT(x),Remm'k 5.2.1(a) If T is a linear operator, thenthat is, DT(x) = T for all x E X.DT(x)t = T(t),(b) If l' = F is a real-valued functional on X; that is, T: X ---t R, and F is Gateaux differentiable at some x EX. thenDT(x)1~ [~ F(x + "t)] .,=() , (7)(5.2.2)and, for each fixed x EX, DF(x)t is a !mear functIOnal of t EX. (c) It may be noted that the Giiteaux derivative is a generalization of the idea of the directional derivative ",-ell known in finite dimensions. Theorem 5.2.1 vided it CXlStS.The Gateaux derivative of an oPl-"1tltor l' is unique pro-P7'fJOf. Let two operators Tt(t) and T 2 (t) satisfy (5.2.1). Then, for every t E X and every 7} > 0, we havE'IIT,(t) _ 1',(1)11 =II (T(X HI; -T(x)-1',(11)-1;(0"-IThenwhere t = (t l ,t2)' DF(OH exists if and only ifIt is clear from this example that the exIstence of the partial derivatives is not a sufficient condition for the existence of the Gateaux derivative. Let X = ROl, F: ROl -4 R, x = (XI, . ,x,,) E fl" and t = (t t2, ... ,t,,) E R". If F has continuous partial derivatives of order 1, " then the Gateaux derivative of F isExample 5.2.4DF(x)' ~k=ol" L: 8F(x) t,. iJ x,(5.2.3)For a fixed a EX, the Gateaux derivative at a,DF(a)'~[tk=l8F(x) ,,]ax,(5.2.4):I'=ais a bounded linear operator on fl" into R" (we know that (R")' = DF(a)t can also be written as the inner productR").DF(a)' = (y, t),wheIT'Y(5.2.5)~ (8F(a) 8F(a) 8F(a)) " ' B " " ' uX" ' !! UXI X2Example 5.2.5 Let X = R", Y = R"', and F = (FI , F2, F3 , , F,.,,) : R!' --t fl'" be Gateaux differentiable at some x E flOl. The Gateaux derivative can be identified with an m x n matrix (atj). If t is the j-th coordinate vector, t = j = (0,0, ... ,1,0, .. ,0), thenlimF(x + 1)t) - F(x) _ DF(x)t = 0,1JQ...... ,'}-'O, to0TJ_ 0 -a,,-,for every i :::: 1,2,3, . . ,m and J :::: 1,2,3, ... ,no This shows t.hat F,'s aE(x) have partial derivatives at x and 0 ' == aij, for every i :::: 1, 2, 3, ... , m and j:::: 1,2,3, ... ,no Th(' Gateaux derivative of F at x has thl" matrix representationx,of, (x)ax, oFm(x) ax,of, (x)8x":::: (a,j).8Fm (x) ax.(5.2.6)This is called the Jacobian matf'ix of F at x. It is clear that if m :::: 1, then the matrix reduces to a row vector which is discussed in Example 5.2.4.Example 5.2.6 Let X:::: C[a,b],Jtl, 11) he a wntinuous real function on [a,b] x [a,b] and g(v,x) be a continuous real function on [a,b] x R withcontinuous partial derivative~~ on [a. b] x R.Suppose that F is an operatordefined on CIa. bl into itself byF(x)(s) "lf{(s, v)g(v, x(v))>lv.(5.2.7)ThenDF(x)h =l+ t)f{(s,v) [:x9(v,X(V))] h(v)>lv.Theorem 5.2.2 (Mean Value Theorem). Suppose tllat the/unehonal F has a Giiteaux tle,'ivative DF(x)t at every point x EX. Then /0'- any points x, x + t EX, there exists a ~ E (0,1) such thatF(x- F(x)~DF(x+ 'I)t.(5.2.8)n",,,. "'_'_""""'''''''10''-''Proof.Put~(o)= F(x+ ot).Then""(0) -J~ [",(o+~t"'(O)]= lim F(xtl..... o+ ot + pt) {3F(x+ ot)~,= DF(x+ at)t.By the Mean Value Theorem for real function of one variable hpplied to we get~(ll - the following notation in the Mean Value Theorem and Taylor's formula: If a and b are two points of a vector space, the notationja,b) ~ {x~ na+ (l-a)bE X/a E [0, I]))a,bj~{x=aa+ (I-a)bE X/a E (0, IIIare used to denote. respectively, the closed and open segments with endpoints a and b. Theorem 5.2.6 (Mean Value Theorem). Let T: J( --+ Y, when; J{ is an open convex set containing a anti h, Y is a normal space anti T/(x)exists for well x Ela,bl -aml T(x) isconti7l1WUSon -[a,b]. Then all. (5.2.15)IIT(b) - T(a)1I"-'......5.4.3Tbe Sobolev Embedding TbeoremsDefinition 5.4.11 (Embedding OrJerot07). Let X and Y be Banach spaces with X ~ Y. The embedding operator j : X --t Y is defined by j(x) =x for all x E X. The embedding X c;: Y is called continuous if and only if j is continuous. that isIIxJly < kllxll xfor all x E X and k is a constant.The embedding X ~ }'" is called compact if and only if j is compact, that is, it is continuous and each bounded sequence {x,,} in X contains a subsequence {X"k} which is convergent in Y More generally, we may define an embedding as an lIlJecuve linear operator j : X --t Y. Since j is injective, we can identify'll. with j(u), in this sense, we write X ~ Y. Let(In l' + t. un') 'J' 1If1!>"o ~ (lnt. Uf,)' dX) 'J'11111", =(5.4,14,(5.4.15)Theorem 5.4.12 Let 0 be a bounded region in R" with n > I. Then (a) The norms 11111t,z and 1I11kz,0 aI'C equitJalerlt. on Hri'z(O). (b) The embedding H1.2(n) (usually derwted by H(O) ~ Lz(O)) iB compact. (c) The embeddingsan, compact.Theorelll 5.4.13 Let n be a bounded region in R", n 2 1 and have Bufficiently smooth boundmy, that. is, an E CO,1 (11 n = 1, then 0 is a bounded open intenJal). Then (a) (Density). COO(O) is dense in H1,'l(O). (b) (Compact Embedding). The embedding H1,Z(n) "-IC",-l (f!),m = 1,2,3, ... , if 11 c R. In R 2 and R3, we have thatH'",2(11) ~ C"'-2(O), m = 2,3 ....(vi) The bounda!)' operator A is vital for t.he formulation of boundary conditiollS in the generalized sense. For example, let f E H1.2(11) and 9 E H 1,2(f2). Then the boundary conditionf=gononis to be undet'stood in the senseAf~Ag m L,(O);that. is. Af(x) = Ag(x) for almost all x E fJl1. For proofs of the above results, we refet to Zeidler [161.5.5Integration in Banach SpacesIn this section, we present somt> important definitions and properties of spaces comprising functions on a real intenal [0, T] into a Banach space X. Such spaces and their properties are of vital importance for studying parabolic differential equations, modeling problf>.1TlS of plasticit)', sandpile growth, superconductivity, and option pricing.Definition 5.5.1 Let (f2,A,/~) be a finite measure space and X a Banach space. u: f2 --t X is called stmngly measuHible if there exists a sequence {un} of simple functions such that lIu.,(w) - u(w)llx --t 0 for almost all w as n --4 00. Definition 5.5.2 (Bochner Integral). Let (n,A,I.t) be a finite measure space, and .t a Banach space. 'Then we define the BocJmrr mtt:.gn.u of a simple function u : n --t X byfor any E E A, where Ci'S are fixed scalars. The Bochner integral of a strongly measurable function u ; n t X is the strong limit (if it exists) of the Bochner integral of an approximating sequence {u,,} of simple functions. Thati~, f Ud,.t =Q.......lElim {U., d/~. 'HoolE, to"-IcL.(O, T;n,is also continuous. (d) Let X" be the dual space of a BanaCh space X, then (L"t 0, Tj .\) the dual of Lp(O, T; X) can be identified with Lp(O, T; X")j that is, we can write (Lp(O, T; Xl)" = L,,(O, T; XO). (e) Proofs of Theorems 5.5.1 and 5.5.2 are on the lines of Solved Problems 2.8.3 and 3.10.3, "-IRepeated applications of this fonnula give5.65.6.1Probl"msSnhrP.(} PmhlemsProblem 5.6.1 If the gradient 'VF(x) of a function F: X --+ R exists and II'VF(x)1I < M for all x E 1(, where I( is a convex subset of.t, then show that!F(u) - F(v)1< Mllv - vIISolution.By Theorem 5.2.2, we have!F(u) - F(v)1 ~IF'ru + .I(v - u))(v - v)1 ~ 1(" F(v + .I(v - v)), (v - v))1 < II" Frv + .I(v - u)lIl1v - vII " Mllv - ullI(as II 'V F(tl) II :0:; M for all A)'U + All E 1(1.'UE[SinceI(is convex; for all1[,vEl(, (I -Problem 5.6.2 Let f . j{J > R possess continuous second partial derivatives with respect to all three variables, and let F : Gl[a,b] --+ R be defined byF(x)~lf(x(t),x'(t.),t)dt.Show that the F'rechet derivative of F, dF(x)h, is given hy["[8f d ax' dF(x)1i =}" ax - lit (8f)] lull+ [8f ]' ax,li CI'n...". "'_'_"_"""10"-"Sol1ttion,F(x+ h) - F(x)l ~ l (~~~-4(f(x(t)+ h(t),x'(t) + h(t),t)- !(xlt),x'(t),t))dt (x(t),x'(t),t))h(t)+ :'~, (x( t), x' (f), t) h(t)) dt + ,'( h, h)where r(h,h)==O(lIhIlC[a,bl)' i.f'"11111 C'la,b] IIr(h, h)10 '"IIhllolo,'l -40dF(x)h~~l [~~(X(t),x'(t),t)h(t)}a.. ;;,(X(t),X'(t),t)h(t)] dtr' [a! _!!- (8!)] hdt .. [Of h]' OX dt OX' OX' "[.(of.after integration by part.Problem 5.6.3X --+ R be a bounded symmetric bilinear form on a. Hilben. space X a.nd J a functional 011 X, often c..alled "energya(,) : XXfunctional", defined byJ(u) == '2a(u.1l) - F(u),Find tl1e Frechet derivative of .J1where FE X*Solution.J("Fbr811arbitrary ,A.. ~EX,+ )~"a(" + ,,, + ) - F(" + )II~ "a(",,,)+ "a(,,,) + "a(",) + "a(,)1II- F(,,) - F()n...", "'_'_""""'''''''10''-''by tl1e bilinearityof a(, .). Using the symmetry of a(,), [a(11,) = a(,u)], we getJ(u+)~ga(u,,,) - F(u)} + (a(",) - F()} +I~,,(,)= J(u) + (alu,,,) - F()} + 2a(,1,IIJ(u + ) - J(u) - {a(",) - F())II1IlIx< ~ MIIlIxlllIx- 2Tl1is implies thatlla(,)1 2 1IlIx1Illxas a(, .) is bounded.. IJI,,+) - J(u) - (a(u,) - F()}I 0 = , \,m 1Illx-+o 1IlIx~dJ(u)= a(",) - F().Since J defined in t.his problem is Frcchet differentiable, it is also Gateaux different.iable and DJ(u) = dJ(u). The derivative of this functional is oft.en used in optimal control problems and variational inp.qualities.Problem 5.6.4 Prove that a linear operator T from a Banach space X into a Banach space 1- is Frechet differentiabl~ if aud only if T is bounded.Solution. Let T be a linear operator and F'n~chet differentiable at a point Then T is continuous (and hen('f' hounded) by Theorem 5.2.4. Conversely, if T is a bounded linear operar.or, then IIT(x + t) - Tx - Ttll = 0, proving t.hat T is Frechet differentiable and T' = T.Problem 5.6.5 [{ depending onProve that for n such thatfE cO'(O),n c R2, there is a constantn",,,. "'_'_""""'''''''10''-''Solution. Let f E Cgo(O). Consider a. rect.angle Q = la, bl x [c,d] as in Figure 5.6.1 with IT c Int Q. Note that f vanishes outside O. Thenf(x,y) ~ , IJyf(x,t)dt foc aU (x,y) E Q.By Hiilder's inequality. we get, 18Integrating over Q, we getProblem 5.6.6 show that[ f'dx I, then prove that embedding H~' (0) C Lz(O) is compact. Problem 5.7.16Prove Green's formula, namely, the equationProblem 5.7.17 Show that the embedding Loo(O, T; X) : Lp(O, T; X) is continuous for all I S p :S 00, Problem 5.7.18Let1 : R --+ Rbe defined byi( u-f:. 0ifu=O'Show l!Jat (a.) if P > I, then1 is SU'lCtly monotone,n...". "'_'_""""'''''''10''--'(b) if P = 2, thenJ is st.rongly monotone.ShOl'''' that for u E L:dO, T: H) andtho'" LPmblem .'i.7.19tE L 2 (0, Tj H),'fi._.. . ~. . . . ~ __,.'"''References[J. Il Adams. Sobolev Spaces. New York: Academic Press, 1975[2) H Cartan. Differential Calculus. Paris: Herman/Kershaw, 1971. [3] PG Ciarlet. Introduction to Numerical Linear Algebra and Optimization. Cambridge: Cambridge University Press. 1989. [4] RF Curtain, AJ Prit.chard. Functional Analysis in Modern Applied Mathematics. London, New York, San F"':aneisco: Academic Press, 1977 [51 R Dautray, .JL Lions. Mathematical Analysis and Numerical tvlethods for Sciencf' and Tffhnology. Vol 2, Berlin-Heidelberg-New York: Springer-Verlag, 1990. [61 L Debnath, P Mikusinksi. Introduction to Hilbert Spaces with Applications. 2nd ed. Sandiego-London-Boston: Academic Press, 1999. [7] J Dieudonne. Foundation of Modern Analysis. New York: Academic Press, 1000. [8j DH Griffe!' Applied Functional Analysis. Chichester: Ellis Horwood Limited Publishers, 1!J81 [9J C\V Groetsch. Elements of Applicable Functional Analysis. New York and Basel: Marcel Dekker, 1980. [1OJ RD Milne. Applied Functional Analysis. Boston-London-Melboume: Pitman Adwmced Publishing Program, 1980. [11] MZ Na"-IChapter 6Optimization Problems6.1 IntroductionThe ha0 0 and 0::; J(u + oow) - J(u), which implies that J(v) > Ju). (2) By Remark 6.2.1(ii), the necessit.y of (6.2.1) holds even without COJlvexity assumption on J. For the sufficienc.y part, we observe thatJ(v) - J(u)::::>:J'(u)(v - u)for every v E K.Since .I is convex,.1((1- 0)11.+ ov) < (1- 0).1(11.)+ oJ (v)for all0E [0,1],J(u+o(v-u))-J(u) J(v)-J(u) >0 'J(v) - J(u)> limJ(u+o(v-u) -J(u)=:.0-.00J'(u)(v - u)> O.This proves that if J'(u)(v - u)> 0, then J has a minimum at u.In.n","'_'_""""'''''''10''-''A functional J defined on a normed space is called coercive if lim00.II x ll_' ooJ(x) =Theorem 6.2.3 (Existence of Solution in R"). Let K bc a nonempty, closed convex subset of Fr' and J : R" --t R a continuous Junction which is coen;ive iJ [( is unboundcd. Then thcre exists at lcast one solutionof(P).Pwof. Lt {ud be a mimmizing sequence of J; t.hat is, a sequence satisfying conditions u~. E J( for every integer k and lim J(U.k) = inf, J(v). k~oo "en This sequence is necessarily bounded, since the functional J is coercive, so that it is possible to find a subsequence {u~.,} whidl converges to an element u E J( (J( being closed). Since J is continuous, J(u) = lim J(Uk')k' ~ -.)inf J(vl whid] proves the existence of a solution of (P). "oKITheorem 6.2.4 (Existence of Solution in Infinite-dimensional Hilbert Space). Lct [( bc (L non-empty, convcx, closed subsd oj a scpamble Hilbcrt space Hand J : H --t R a convex, continuous functional which is cocrcivc iJ [( is unbounded. TIlen (P) has at least one solution.Proof. As in thl' previous theorem, K must bP bounded under the hypotheses of the theorem. Let {Uk} be a minimizing sequence in [(. Then by Theorem 4.4.7, Iud has a weakly convergent subsequence Uk' ----" u. By Corollary 4.4.1 J(u) < liminf J(Uk')' Uk' ----" u which, in turn, shows that U is a solution of (P). it only remains to show that the weak limit u of the sequence {Uk'} belongs to the set J0for every integer fThe weak convergence of the sequence {w,} to the element u implies thato < h =(Pu limu. w( - Pul = (Pu - u.u - Pu) = -jfu - PuII 2::;O.Thus,Pu=uanduEK.IB.......Remark 6.2.2(i) Theorem 6.2.4 remallls valid for reflexive Banach, to-Lagrangc cquationiJF d (iJF) =0 au-dx au'in [a, b] wtth tJ,e boundary condition u(ll) = PmofJ(uLet u(a) = 0 and u(b) = 0, {hen=Q(6.3.2)llnd u(b) = (3.+ av) ~ J(u)it> [F(x, u.J.... av, u ' + OV') " ')F(x, U, U') d.r.I(6.3.3)Usmg the Taylor series expansIOnF(.x,u+ttV,U +(11) = F(x,u,u +fi vau +V au'(iJF ,OF)+ 2f v an0'(aF aF)' + ... ,+V'{}II.'B..... ,, to0O(6.4.4)n.n". "'_'_""""'''''''10''-''is etltirely contaitled within the ball and converges to a zero of F in S,,(uo)which is 'Utli(JUe. Furtbenllore< ,. II Uk _ U II - II", - "Oil, 1-,'(6.4.5)Theorem 6.4.2 Let X he a Banach S])(lce, U an open subset of X,F : V c X --t V, and V a normed space. Furthermore, let F be contimwilsly differetltiable over U. SU]J]JOse that u is a point of V such that F(u)=:0, A"", P(u) : X --t-J'boundnilinear amI bijectiVe} ,and),{ sup IIA k k?OAII.u(x,l']S IIAtil.u(1',X]< 1(2Then there exists a closed ball, Sr(u), with Cetlter u arul rudi1l$ r SItch thal for every point Uo E Sr(U), the seqnence {uk} (/efineJi by k1S> 0,(6.4.6)conl.ame(l tTl Srlu), and Cf)nverges 1.0 a ]JOmt u, which IS the only zero of F in the ball Sr(u). FUrtheJYTlore, there exists a number")' such thatk> O.(6.4.7)As a couscquem::e of Theorem 6.4.1, we get the following result:Corollary 6.4.1 Let L' be an open sltbstd vf a Banach S])(J.ce X aud F : V C X -4 R which is twicf' (/ijJerentiable in the O]Jen set U. SUIJIJose that there are three con.~lants: 0:. B." .~1tch that Q > 0 awl S,,(uo) =: E{vXlllv-uollawlsa} C V, Ak(v) E S[X, X"]ll1ul bijective for every v E So:(u)Then the seqltetlce {ud (/efitlC(/ byUk-t-1 "'" Uk - A;l (Uk' )F'(Uk),k>k'>O - -n.n","'_'_""""'''''''10''-''is cotltained in the ball Sa(UO) and cotwerges to a zero of F ' , say u, which is the otlly zero in this ball. Furthermore,< II Uk _ UII - II"1 - "oil, l' '}.As a consequence of Theorem 6.4.2, \\'e get the following result:Corollary 6.4.2 Let L' be au allen sltbset of a Banach slmcr. X and F : U C X --+ R a flttlction which is twice difJeretltiable in U Moreover, let U be a lwint of U such thatF/(u) = 0, F"(u) E BIX,X*] and bqe.chve aml.\ >'dlwll 2 for allu, w Eeigell\~.llue ofR",where>'.denotes the smallestA.n.n". "'_'_""""'''''''10''-''- (y,v), A; R" -J' (R"r = R n . Since \7J(ud and \7J(Uk_H) are orthogonal and \7J(v) = Av - y, we have(ii) Let J(v) ==~(AV,v)(\7 J(uk+d, \7 J(ukl) == (A(Uk - op(Uk)(Auk - y)) - y, AUk 2Y) = o.IIwkil where Wk = AUk - Y = \7J(uk). A (AWk,Wk) single iteration of the method is done as follows;This implies that op(ukl ==(i) calculate vectorWI;= AUk - Y;(H) calculate the number 0,Iii) II \7 F( 0) - \7 F( v) II < Plio - v II loreo',," v, v E R".Furthennore, let there exist two numbers a and b SItch tJtat20 o < a < '1 and >." are, respectively, th" least and the largest eigenvalues of the symmetric positive definite matrix A.n.n", '"'_._""""'''''''10''-''Proof of Thcorell1 6.4.1.First, we prove that for every integer k 2: I,~ ~IIF(",_>lII,II", - a,_>l1lIuk - noll< u eqUIvalently Uk E Su(uo)~ ~lia,IIF(uj' )F(a,,_dll"-ITherefore, we gel-1.ro ..6.uvdx ="J&n&v L ro - - d x . ox- oxi=l..(7.4.7)By Equations (7.4.6) and (7.4.7), we obtain Equation (7.4.3) where a(',') and L are given by Equations (7.4.4) and (7.4.5), respectively. Existence of the sohltion of Equation (7.4.3): Sincea(u,v) = '\' n " " d x L uX I' uX t' .=1= '"'"1.&u&V"Lt=l"1.&u&u = a(v,u), ~~dx nuX uXa(, .) is symmetric.a(u,u):::"1.&u&u L ro ux. uX, ~~dx1=1 ..2':kllltlllfl~(fl)(by Theorem 5.4.8,S("ealso Remark 5.4.3 ).Thus, a(,) is coercive. We haveL(v]+ V2)= kf(VI+ v2)dx= kfVldX+ kfV2dX=L(Vd+L(V2)L()'v) =1.!(),v)dx = ),1.!v(dx)= ),L(v),where), is a scalar. Also,by the CSB inequalit". Since>I'!EL,(rl) 11111 = (1.lfI'dx)S k,k>O ,Q...... ,, to"-IDefinition 7.5.1 For gi ven y E 11, Equation (7.5.1) is called uniquely approximation-solvable if the following conclitions hold:(a) Equation (7.5.1) ha 0:IIull 2 fo" all u E H and fIXed a > O. Under the hypotheses for 0: > 0,all" - unll~l17'un- yll(7.5.6)(d) 7' = S + U. where S : H -+ Ii is linear continuous and coercive and U is linear and compact, and Tu = 0 l77lplies u = O. In cases (a) and (c), 111 = 1. In case (b), m is independent of y. We prove here cases (a) and (c) and refer to Zeidler [15} for the other cases.OlPnllPmof.(a) Since P"u" = u" for u" E 11" and IIP,ISIl < IISI1 < 1, = 1 by Theorem 3.4.1(3)). The following equationsu+ Su= y,u E If,(7.5.7)Q......., toOl'._......... ",,"l1>"-Iand (7.5.8) have unique solutions by the Banach Contraction Mappin}!; Theorem (Theorem 1.2.1). F\uthennore,II(! + p"S)-'11 ~I)p"S)' ~.1:=0=L 11511'k=O=By (7.5.7) and (7.5.8), (1 Hence,+ P"S)(u -'U,,) =U -p"u.lIu - u"1I ~ (I -1I511-')lIu - P"ull ~ (1 -IISII-')d(u, p"u).(c) For all u Eli, we hm'e Pnu =Uand hence(P"Tu,u) ~ (Tu, p"u)> ollull'.(7.5.9)Thus, the operator PnT . Ii" --t fin is strongly monotone. By the Laxl\'lilgram Lemma, the two operator Equations (7.5.2) and (7.5.3) have unique solutions. If n > j, then it follows from (7.5.3) that(Tu",Wj) = (y,Wj),(TUn, u.,) = (y, 'Un).(7.5. to) (7.5.11)By (7.5.11),This yields a pTlon estimatecllu.. 11~lIyll,that is, {Un} is bounded.U"Let {u ll '} be a weakly convergent subsequcnCf' with 'Un' -->. V as n --t 00 (Theorem 4.4.7). By (7.5.10), (Tull"w) --t (y,w) as n --t 00 for all W E lin. Since H II is dense in Ii and {Tull} is bounded, we obtainU"TUn-->.Y as n --t00.(Theorem 4.4.7).n...... "'_'_""""'''''''10''-''Since T is linear and continuousTu",->.Tv as n --)00.(Theorem 4.4.1(i)).Hence, Tv = y; that is, v = u. Since tlte weak limit u is the same for all weakly convergent subsequences of {u,,}, we get Un U as n --t 00. It follows from--->.allu" -UJj2< (T(u"as n- u),u" - u)-= (y, u,,) - (TUn, u) - (Tu, Un--t 00;u) --t Uthat istI"--ttIas1l--t 00.Therefore,ollu" - ull'and Tu = y. Therefore,~IITu" - Tuliliun - ull,allu" - ull < IITu" - yll.I7.5.2R11,vleigIJ-llitz-Galerkln MetbodThe Rayleigh-Ritz-Galerkin method deals with the approximate solution of (7.5.1) in the fonn of a finite seriesUm=LJ=!mCjj --I-rPo,(7.5.12)and its weak formulation (variation formulation), a(u, v) = F(v), where the coefficients cJ ' called the Rayleigh-Ritz-Galerkin coefficients, are chosen such that the abstract variational formulation a(v, w) = F(v) holds for v =;,i:=: 1.2..... m: that is,i== 1,2, ...,1H.(7.5.13)Since a(, .) is bilinear, (7.5.13) takes the (ormL: a(" ')', ~ F(;) j=1OJma(" OJ,(7.5.14)Ac- b, -(7.5.15)..... ,Q . ,>0' ::j;. ).,." 11 == 1,2,3, ... , Problem (7.6.4) has exactly one weak soluuon for every y E 2(0). If the eigenfunctions lPj(x) belong to the domain Dr of the operator T, then the relat.ion for e\wy v E H gives the classical eigenvalue problem(7.6.9)If WI'! choose {lPj} as tlw basis of 110 "(0). then for t.he N-paramet.rk Ritz soludon of Tu == Y, we writeNUN (x)==L>jlPj,j='where cj =:: (Y,lPj)' Thus, eigenfunctions associated with the eigenvalue problem Tu == AU can be use"-IExample 7.7.10 (The Timelndependent Schrodinger Equation in Quantum Mechanics).I' -'-tl1j.J+(E-F)1j.J=O, 2m(7.7.16)where m is the mass of the particle whose wave function is 1/', Ii is the universal Planck's constant, F is the potential energy and E is a constant. If l! = 0, (7.7.16) reduces to the Helmholtz equation. We prove here the existence of the solution of Stokes equation and refer to Chipot [1], Dp.hnath and lvlikusinski [4], Quarteroni and Valli [8], R.eddy [9], Reddy [10], and Siddiqi [12] for variational formulation and existence of solutions of other boundar)' value problems. A comprehensive prcsentation of variat.ional formulation and existence of solution of parabolic equations including classical heat equat.ion is given in Chipot [I, Chapters II and 12J. Existence of the Solution of Stokes Equations. We have-Iltlu+ grad p-f= 0in u E 0 in u E 0 on r,(Ul' U2, U3)(7.7.17) (7.7.18) (7.7.19)divu=O u=Owhere f E L 2 (O) is the body force vector, u ::: vector, p is the pressure. and It is the viscosit)'. We introduce the following spaces:D ~ {u E C;:'(fI)1 dlv u ~ OJHis the velocity= {u E HJ(O)x HJ(O)/ dh' u= O}(7.7.20)Q~ {PE L,(O)I llxlX~O}The space H is eqU1pped with the inner product{v, u)u =t: IngradVi'gradUi dx ,(7.7.21)where n is the dimension of the domain U C IR". The weak formulation of Equations (7.7.17) (7.7.19) is obtiuned using the familiar procedure (Le., multiply each equation with a test function and appl)'ing Green's formula for integration by parts (Theorem 5.4.7).n.n","'_'_""""'''''''10''-''We obtain for v E V,(-I-I.6.u+ grad p - J,v> = 0or /1. ~Nlgrad u, . grad v,dx = (grad p, v)= (1,v)+ (1, v)11for everyE V.As for v E V, we ha\'e div v = 0 and v = 0 ona(v, u)r, giving= (v, f)for ever)' v E V,wherea(v, u) =t lit;=1gradVigrad u,dx.(7.7.22)nWe now have the following weak problem: find u E I-l such thata(v, u)~(v, f)(7.7.23)holds for every v E 11. The proofthat the weak solution of Equation (7,7,23) is the classical solution of Equations (7.7.17) - (7.7.19) follows from the argument (see Temam[14])for every v E V. (7.7.24) This does f10t imply that -It.6.u - J = 0 because v is subjected to the constraint (because v E H) div v = O. Instead, Equation (7.7,24) implies that-1-1.6.11 - / = -grad p,(7.7.25)because (necessary and sufficient)(v, grad p)= (div v,p) = 0for every p E Q.(7.7.26)The bilinear fOHn in Equation (7.7.23) satisfies the conditions of the Laxl'vlilgram Lemma. The continuity follows from Equation (7.7.22) using the CBS inequalityla(v,u)1 =l; LIt grad Vi' grad ujlix""[t,1,~[g,adV'['dx] 'I'[t, 1,[g,adU'['dX] 'I','lIvll H lIu[,"-n.n","'_'_""""'''''''10''-''The F-ellipticit:y of a(',') follows fromla(v, v)1 = 1-1LN,==1 n1grad v, . gr"-Itogether with the corresponding approximation equationUnEX,,,n= 1,2,3, ..(7.8.3)Let us make the following assumptions: (i) X n is a subspace of the Banach space X and dim.\n = X ... is a projection operator onto X n . (ii) The operators 5 : X and 1+5; X -+ X-~ i~Fl.,Pn ; X -+X and 5 n : ...\11 -~ X n are linear and bounded,onto.(iii) There exists a constant lin with d(Su. X ... ) ~t3nllullfor allu EX.(iv) As n -~ 00, IlPnS - Snllxn -~ 0, II P., 1It3'1 -~ 0 and IIP"lId(y,X,,) -~ o for all y E X hold. Show that Equation (7.8.3) is uniquely approximation-sol\"-IIf IIi/. - 11'hJl S CJ{' for 0: > 0 where C is a positive constant independent of 11 and '/.I'h and 11 is the characteristic length of an element, then 0: is called the rate of convergence. It may be observed that the convergence is related to the norm under consideration. say, L.-norm. energy norm (L 2 -norm) or Loa norm (ur Slip norm).Corollary 8.2.1 Suppose there e:asts a dense subspace U of H and a mappinf} Th ; H --t fh such that lim IIv - Thvll = 0 V v E U. Then,~OPmof. Let e > O. We choose v E U such that for C > 0, lIu-vll < e/2C (U is dense in 1-1) and It sufficiently small such that IIv - T"YJII S e/2 C (This choice is Vossible in \iew uf the hyvot.hesis). Then by Theorem 8.2.1, and in view of these relationslIu-tlhll < Cw,EH"inflIu-vIlIlbecause T"V E HI,< II"!! - Thvll,S C [1111. -vII + lit! -TIlVU]Ge= e.Ce< 2C + 2CTherefore,IThere is a vast literature concerning convergence for special types of norms and spaces and special types of boundary and initial value problems. For interested readerfl, we refer t.o \Vahlbin (44]. Ciarlet [191, Ciarlet ''-I..... ,2. The baryc.entric coordinates are affine functions of XI, xz, .. . ,xn ; i.e., they belong to the space PI (I,he space of all polynomials of degree I).A,:::= Lb;jxj +bin+i.j=1"I< i < n + 1,(8.2.21)where the matrix B = (b,j) is the imrerse of matrix A. 3. TllP barycpntrk or center of gravity of 1m n-simplex J( is I,hllt point of !< all of whose barycentric C'.oordinal.es are e"-ISince {Wi}:S,j is a basis of fh, the solutionNth)Uh=L"(kW~"~/k(8.3.14) are the solutions of theJ'=Iof Equation (8.3.13) is such that the coefficlems following linear system:N(h)La(wl;,l/Jthk = L(wt}for 1 ::;e< N(h),(8.3.15)k=1where (8.3.16)and(8.3.17) Thus, if we know the stiffness matrix a(w~",wt) and the load vector L(wd, "(k can be determined from Equation (8.3.15) and pUHing these values in Equatiun (8.3.14), the solution of Equation (8.3.13) can be calculated.Practkal Method to Compute a(wj , w,). We havea(u,v) ==L(Vu Vv\- uv)dxL r (Vll' Vv + uv)dx.KET..10In the elementJ(,we can write,llo= 1u(x) - " U m"II' >'0 (x) L..J K,V(X) = " Lp=lllml.'j{>'p (x), Kwhere (m o K)a=1.2,3 denotes the thfef' vertices of the element K and >.~ (x) the associated barycentric coordinates. Then Equation (8.3.18) can ben...... '"'_'_""""'''''''10''-''rewritten aswhereK ao{3 =J.K(y>.K . y>.K a {3+ ),K>.K)dx' o{3The matrix A K -= (a:;'{3)I:'So9.1:'S{39 is called the element :stiffne:s:s matru of K.8.4Basic Ingredients of Boundmy Element MethodFirst, we discuss weighted residual methods, in verse problem and boundary solutions. The boundary element method is explained with the help of Laplace equation with Dirichlet and Neumann boundary conditions. For more details such as error estimation, coupling of finite clement and boundary element methods and applications of this method to parabolic and hyperbolic equations, we refer to {I, 4-13,18,21,24,37,40).8.4.1lVeighted Residuals l\JetlJOdIn Section 7.2, we have seen that a boundary \'"-IEquation (8.4.35) takes the form2u , + L1N _Hi,Uj=L,=1NG i , qj.(8.4.37)j=lHere, the lIuegraJs in (8.4.36) are simple and can be evaluated analytically but, in general, numerical techniques will be employed. Let us defineH"H ij =ifi-j{-, Hi' + 2"N(8.4.38)ifi=j,then (8.4.37) can be written asNj=lLHijUj = Lj=lG;j qj,whieh can be written in the form of matrix equation asAX = F,(8.4.39)where X is the vector of unknown u's and q's (Either u's or q's will be known and X has N components), and A is a matrix of order N Thus, all the values of potentials and fluxes on the boundary are knuwn and une could calculate tllP vaJues of potentials and fluxes at any interior point usingUj=l=qu dVr-lN,=1q*Udr.(8.4.40)Equation (8.4.40) represents the integral relationship between an internal point, the boundary values of U and q and its discretized form isNU.LQ,G1j - Lu,H ijj=1.(8.4.41)The values of internal fluxes can bf' determined by differentiating (8.4.40) which gi ves us:~ = -llwhereXIq U - "-I9.1.2Variational Inequalities in Social, Financial and l\lanagement SciencesIn 1980, Cottle, IooI"_""""'''''''IO''-I9.2.2Formulation of a Few Problems in Terms of Variational InequalitiesR be a differentiable real-ntlued function defined on the closed interval of R. We indicate here that finding the point Xo E J = la, bl suchthat1. Minimization of a Single-valued Real}""'unction: Let f; [a, b] -----tf(xo) ~ f(x) for all x E J; that is, f(xo) = inf f(x), . E Jis equivalent to finding the solution of a variational inequality. WE' know that (a) if a< Xo < b, then1'(xu) =u(b) if Xo = a, then 1'(xo)? 0; and (c) if Xo:= b, then 1'(xo) < o.From this, we see that(f'(xo),x - xo)> 0 for all xE[a,bl,xo E [a,b);thaI is, Xo is a solution of Modell, where F :::: O,a(u,v - u) (f'(u), v - u) :::: (Bu, v - u), B :::: l' : R -----t n lineal' or Xo is solution of Model 2, where F = 0 and T = 1'.2. Minimization of a Function ofVariables: Let f be a differentiable real valued function defined on the closed convex set K of R". We shall show that finding the minima of f, that, is searching the point Xo E K such that f(xo) = j~Lf(x), that is, f(xo) < f(x),11for all x E l( is equivalent to finding the point Xo E K, which is a solution of MadeliaI' Model 2. Let Xo E K be a point where the minimum is achieved and let x E K. Sin~ K is convex, the segment (1 - f.)xo + tx = Xo + t(x - xo),O < t < I, lies in K. /(t) = f(xo + t(x - xo),O < t < 1, attains its minimum at t = 0 as in (1). 1"(0)=:(grad f(xo), (x - xo) ? 0 for any x E= \1f(x) =8J ( ~'~""'~ ,x VXI VX2 vX"OfOf)1(,where gradf= (XI,X", . . ,x,,) E IC This.shows that, Xo E J( is a solution of tht=> variational inequality, (gradf(xo), (x - xo? 0 for any x E 1(, or Xo solves Model 2 wheren",,,. "'_'_""""'''''''10''-''u = xo, v = x, T = grad 1 = \71, F = O. Thus, if 1 is a convex function and u E K is a solution of variational inequalit.y (Model 2) for T = grad 1 = \11, thenf(x)~vEKinf f(v)3. Consid(lr UlP following boundary value problf'm (BVP) - dx' ~ f(x) on (0,1)u(O)d'u(i)(H)> 0;u(l)>0du (-f ),~" 0 for all v E K. Now if we choose v = 2u, which is permissible since K is a cone, we obtain (Qu,u) > 0, By choosing v = 0, we have (Qu,u) < O. By combining these two inequalities, we obtain (Qu, u) = 0 and, finally adding this to (Qu, v - u) > 0, we &'et the desired result. \Ve show now that Model I, where Hand [( are as chosen above,a(u.v) = (Bu, v) =Q.......l'dudv x and (F,v) = --d o dxdxl' l(x)v(x)dx, to io j(x)v(x)dxJ(x)u(x)dxr'(9.2.16) (9.2.17)l' (:)' ldx =Assume tlUlt the solution u of (9.2.16) is smoother than stnct.ly required for it to be in HI(O,I), for example u E C 2 (0, 1) and 11 E Cg"(O, 1). Integrating by parts the left-hand side of (9.2.16), we find10>1 - : v(x)dx11+ [( ~:) x=1v(l) - ( : ) x=o V(O)] dx(9.2.18)j(x)v(x)dxSince v E [( n CO'(O, 1) and (9.2.16) also hold for (-v), we conclude thatt iodudv (I dx dx dx < j(x)v(x)dx.io(9.~.19)This together with (9.2.16) implies, 1dudv --dx:::: o dxdxl'0j(x)v(x)dx.(9.2.20)An integration by parts here yields (9.2.17) and, as v E Cg"(O.I); that is, v(O) :::: v(I), the boundary term ''8nishes and we obtain-~~ :::: lex) in (0,1).In view of (9.2.17), (9.2.18) gives us(d")dxx:o::lv(l) -(d")dx x=o v(O)> 0, for u, vEl(.(9.2.21 )Q...... ,, to 0, i:::: 1,2, .... n}.R+,R+ : :R+Proof fG-, soLet u Le a ~olution to the NCP. Then (F(u), v} ::::-: 0 't:/ v E(F(u),v - u) = (F(u),v) - (F(u),u) :::: (F(u),v)>0because (F(u),u):::: O.Thus u is a solution of the variational inequality. To prove the converse, let u be a solution of the variational inequality. Then v :::: U + Ci, ; :::: (0,0, ... ,1,0, ... ) (1 in the i-th place) is an element of R~'_, so 0 5 (F(u),u + tot - u) :::: (F(u),;) :::: Fi(u) or F(u) E R+-. TllU~, since v:::: 0 E R+, (F(u),n) 5 O. But u,F(lI) E R~_ implies that (F(u),u) > O. Hence, (F(u),u) :::: 0; that is u is a I solution of the NCP.Q......., to"-I~0,q E ](1')is the mathematical fommlalion of the situation "supply equals demand". This ideal situation cannot always be realized. We are therefore satisfied with a weaker condition. First, it is rea.sonable to assume that{p.q}> O.for all pE lvi, q EK(p).This is the so-called the law of Walras. Roughly, it means that. we only consider economical situations with a supply excess. We call (j5, ij) with ij E K(p) a Walms equilibrium if and only if the following holds. 0 < (p, q) < (p, lj) for all p E Ai. Broadly speaking, this means that the '~d.lue difference between the supply and demand becomes minimal. The vector p is called the equilibrium price system. The fundamental problem of mathematical economics is to find conditions for the supply excess map !( which ensures the existence of the Walras equilibrium. This is equivalent to finding the solution of Model 6, where H := R", AI = unit cube, {Tu, v} = (u, v) inner product in R". Moscu (22] has given a theorem dealing with th~ solutioll of this model9.3Elliptic VariatiolJalllJequalitiesLet H be a Hilbert space and K a non-empty dosed convex subset of H. Further, let a(,) be a bilinear form on H x H into Rand F E H" which is identified with element, say F", or y of If, by the Reisz representation theorem. Then the Modell (variational inequality problem: VIP, for short) takes the form: Find u E [( such thata(u,v - u)0'> F(v - u):= {y,v - u} for all v E [((9.3.1 )(Au, v - u)> F(v -u) = (F, v - u) = {y, v - u},~(9.3.2)whe"e A, H ---> W, (Au, v) ~a(u,v),IIAIl in l\,todel 2, we find u E K such that{Tu,v - u}lIali.[(,> (F,v - u) "Iv E(9.3.3)where T ; H - - t H" ur mtu itself in the c~ of real Hilbert space. [n Section 9.3.], we shall study the existence of solution of (9.3.1) when a(',') is Ixmnded and coercive bilinear form (not necessarily symmetric),n...... "'_'_""""'''''''10''-''More precisely, we present the celebrated Lions-Stampacchia theorem. Prior to this study, it will be proved that, for the bilinear bounded coercive and symmetric bilinear form, the optimization problem for the energy functional is equivalent to a variational inequality problem, that. is, "Find u E [( SUdl that J(u)~J(v) 'r/v E IC where J(v) = 2u(v.v) - F(v) is equivalent to1(9.3.1)". Section 9.3.2 is devoted to the existence theorem for the VIP (9.3.3) along with the Wtz-Galerkin and Penalty methods.9.3.1LionsStampacchia ThooremFor a symmetric bilinear form, the Lions-StampacdJia theorem follows (rom Theorem